Finite-Support Polynomial Operations, Zero-Divisor Degree Collapse and an Infinite Carrier
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batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 3, 0, 0]. Give (raw,zero,normalized,degree).
View answer and explanation
([1, 3, 0, 0], 0, [1, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 0, 0], 0, [1, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
2. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 0, 0, 0]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 0, 0, 0], 0, [3], 0)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 0, 0, 0], 0, [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
3. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 0, 0]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 2, 0, 0], 0, [3, 2], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 0, 0], 0, [3, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
4. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[0, 3, 0, 0]. Give (raw,zero,normalized,degree).
View answer and explanation
([0, 3, 0, 0], 0, [0, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([0, 3, 0, 0], 0, [0, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
5. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 0, 0]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 2, 0, 0], 0, [3, 2], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 0, 0], 0, [3, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
6. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 0, 0, 0]. Give (raw,zero,normalized,degree).
View answer and explanation
([2, 0, 0, 0], 0, [2], 0)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 0, 0, 0], 0, [2], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
7. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 0, 1, 1]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 0, 1, 1], 1, [3, 0], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 0, 1, 1], 1, [3, 0], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
8. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 3, 1, 1]. Give (raw,zero,normalized,degree).
View answer and explanation
([1, 3, 1, 1], 1, [1, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 1, 1], 1, [1, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
9. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 1, 1]. Give (raw,zero,normalized,degree).
View answer and explanation
([2, 3, 1, 1], 1, [2, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 1, 1], 1, [2, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
10. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 1, 1]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 1, 1, 1], 1, [3], 0)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 1, 1], 1, [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
11. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 1, 1]. Give (raw,zero,normalized,degree).
View answer and explanation
([2, 3, 1, 1], 1, [2, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 1, 1], 1, [2, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
12. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 1, 1]. Give (raw,zero,normalized,degree).
View answer and explanation
([1, 2, 1, 1], 1, [1, 2], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 1, 1], 1, [1, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
13. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[0, 3, 2, 2]. Give (raw,zero,normalized,degree).
View answer and explanation
([0, 3, 2, 2], 2, [0, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([0, 3, 2, 2], 2, [0, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
14. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 2, 2]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 2, 2, 2], 2, [3], 0)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 2, 2], 2, [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
15. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 2, 2]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 1, 2, 2], 2, [3, 1], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 2, 2], 2, [3, 1], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
16. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 2, 2]. Give (raw,zero,normalized,degree).
View answer and explanation
([2, 3, 2, 2], 2, [2, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 2, 2], 2, [2, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
17. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 2, 2]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 1, 2, 2], 2, [3, 1], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 2, 2], 2, [3, 1], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
18. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 2, 2]. Give (raw,zero,normalized,degree).
View answer and explanation
([1, 2, 2, 2], 2, [1], 0)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 2, 2], 2, [1], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
19. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 0, 3, 3]. Give (raw,zero,normalized,degree).
View answer and explanation
([2, 0, 3, 3], 3, [2, 0], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 0, 3, 3], 3, [2, 0], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
20. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 3, 3]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 2, 3, 3], 3, [3, 2], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 3, 3], 3, [3, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
21. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 3, 3]. Give (raw,zero,normalized,degree).
View answer and explanation
([1, 2, 3, 3], 3, [1, 2], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 3, 3], 3, [1, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
22. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 3, 3]. Give (raw,zero,normalized,degree).
View answer and explanation
([2, 3, 3, 3], 3, [2], 0)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 3, 3], 3, [2], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
23. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 3, 3]. Give (raw,zero,normalized,degree).
View answer and explanation
([1, 2, 3, 3], 3, [1, 2], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 3, 3], 3, [1, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
24. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 3, 3]. Give (raw,zero,normalized,degree).
View answer and explanation
([3, 1, 3, 3], 3, [3, 1], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 3, 3], 3, [3, 1], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
25. Normalize a Finite-Support Coefficient Array
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 3, 0, 0, 0]. Give (raw,zero,normalized,degree).
View answer and explanation
([1, 3, 0, 0, 0], 0, [1, 3], 1)
Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 0, 0, 0], 0, [1, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate
26. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0], right=[2, 1, 0, 0]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([1, 3], [2, 1], [3], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [2, 1], [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
27. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 0, 0], right=[2, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3], [2, 3], [1, 3], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [2, 3], [1, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
28. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[0, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3, 2], [0, 3], [3], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [0, 3], [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
29. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([0, 3], [2], [2, 3], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([0, 3], [2], [2, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
30. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3, 2], [1, 3], [2], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [1, 3], [2], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
31. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 0, 0], right=[1, 2, 0, 0]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([2], [1, 2], [3, 2], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2], [1, 2], [3, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
32. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 1, 1], right=[1, 3, 1, 1]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3, 0], [1, 3], [3], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 0], [1, 3], [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
33. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 1, 1], right=[0, 1, 1, 1]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([1, 3], [0], [0, 3], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [0], [0, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
34. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[0, 2, 1, 1]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([2, 3], [0, 2], [3], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [0, 2], [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
35. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 1, 1], right=[0, 3, 1, 1]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3], [0, 3], [2, 3], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [0, 3], [2, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
36. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[1, 2, 1, 1]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([2, 3], [1, 2], [2], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [1, 2], [2], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
37. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 1, 1], right=[3, 1, 1, 1]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([1, 2], [3], [3, 2], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [3], [3, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
38. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 2, 2], right=[1, 0, 2, 2]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([0, 3], [1, 0], [3], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([0, 3], [1, 0], [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
39. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 2, 2], right=[1, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3], [1, 3], [0, 3], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [1, 3], [0, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
40. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[2, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3, 1], [2, 3], [3], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [2, 3], [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
41. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 2, 2], right=[1, 2, 2, 2]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([2, 3], [1], [1, 3], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [1], [1, 3], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
42. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[0, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3, 1], [0, 3], [1], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [0, 3], [1], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
43. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 2, 2], right=[0, 1, 2, 2]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([1], [0, 1], [3, 1], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1], [0, 1], [3, 1], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
44. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 3, 3], right=[3, 2, 3, 3]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([2, 0], [3, 2], [2], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 0], [3, 2], [2], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
45. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 3, 3], right=[0, 3, 3, 3]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3, 2], [0], [0, 2], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [0], [0, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
46. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[0, 1, 3, 3]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([1, 2], [0, 1], [2], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [0, 1], [2], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
47. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 3, 3], right=[0, 2, 3, 3]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([2], [0, 2], [1, 2], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2], [0, 2], [1, 2], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
48. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[3, 1, 3, 3]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([1, 2], [3, 1], [1], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [3, 1], [1], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
49. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 3, 3], right=[2, 3, 3, 3]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([3, 1], [2], [2, 1], 1)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [2], [2, 1], 1).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
50. Add Polynomial Coefficients with the Parent Ring Table
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0, 0], right=[2, 1, 0, 0, 0]. Give (normalized_left,normalized_right,sum,degree).
View answer and explanation
([1, 3], [2, 1], [3], 0)
Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [2, 1], [3], 0).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate
51. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0], right=[2, 1, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
52. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 0, 0], right=[2, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3], [2, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3], [2, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
53. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[0, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3, 2], [0, 3], [0, 2, 3], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 2], [0, 3], [0, 2, 3], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
54. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([0, 3], [2], [0, 3], 1, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([0, 3], [2], [0, 3], 1, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
55. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3, 2], [1, 3], [1, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 2], [1, 3], [1, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
56. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 0, 0], right=[1, 2, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([2], [1, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2], [1, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
57. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 1, 1], right=[1, 3, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3, 0], [1, 3], [1, 0, 3], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 0], [1, 3], [1, 0, 3], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
58. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 1, 1], right=[0, 1, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([1, 3], [0], [1, 3], 1, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 3], [0], [1, 3], 1, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
59. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[0, 2, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([2, 3], [0, 2], [0, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 3], [0, 2], [0, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
60. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 1, 1], right=[0, 3, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3], [0, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3], [0, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
61. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[1, 2, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([2, 3], [1, 2], [1, 3, 2], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 3], [1, 2], [1, 3, 2], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
62. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 1, 1], right=[3, 1, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([1, 2], [3], [1, 2], 1, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 2], [3], [1, 2], 1, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
63. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 2, 2], right=[1, 0, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([0, 3], [1, 0], [1, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([0, 3], [1, 0], [1, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
64. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 2, 2], right=[1, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3], [1, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3], [1, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
65. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[2, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3, 1], [2, 3], [2, 1, 3], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 1], [2, 3], [2, 1, 3], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
66. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 2, 2], right=[1, 2, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([2, 3], [1], [2, 3], 1, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 3], [1], [2, 3], 1, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
67. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[0, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3, 1], [0, 3], [0, 1, 1], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 1], [0, 3], [0, 1, 1], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
68. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 2, 2], right=[0, 1, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([1], [0, 1], [1], 0, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1], [0, 1], [1], 0, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
69. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 3, 3], right=[3, 2, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([2, 0], [3, 2], [3, 0, 2], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 0], [3, 2], [3, 0, 2], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
70. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 3, 3], right=[0, 3, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3, 2], [0], [3, 2], 1, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 2], [0], [3, 2], 1, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
71. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[0, 1, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([1, 2], [0, 1], [0, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 2], [0, 1], [0, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
72. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 3, 3], right=[0, 2, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([2], [0, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2], [0, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
73. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[3, 1, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([1, 2], [3, 1], [3, 2, 1], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 2], [3, 1], [3, 2, 1], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
74. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 3, 3], right=[2, 3, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
75. Finite Convolution May Lose Degree over a Ring with Zero Divisors
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0, 0], right=[2, 1, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
View answer and explanation
([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)
Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate
76. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=1. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(1, [0, 1], 2, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (1, [0, 1], 2, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
77. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=2. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(2, [0, 0, 1], 3, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (2, [0, 0, 1], 3, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
78. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=3. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(3, [0, 0, 0, 2], 4, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (3, [0, 0, 0, 2], 4, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
79. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=4. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(4, [0, 0, 0, 0, 2], 5, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (4, [0, 0, 0, 0, 2], 5, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
80. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=5. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(5, [0, 0, 0, 0, 0, 3], 6, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (5, [0, 0, 0, 0, 0, 3], 6, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
81. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=6. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(6, [0, 0, 0, 0, 0, 0, 3], 7, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 0, 0, 3], 7, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
82. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=7. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(7, [1, 1, 1, 1, 1, 1, 1, 0], 8, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (7, [1, 1, 1, 1, 1, 1, 1, 0], 8, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
83. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=8. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
84. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=9. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
85. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=10. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
86. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=11. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
87. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=12. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
88. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=13. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
89. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=14. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
90. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=15. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
91. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=16. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
92. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=17. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
93. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=18. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
94. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=19. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
95. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=20. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
96. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=21. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
97. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=22. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
98. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=23. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
99. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=24. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
100. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=25. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).
View answer and explanation
(25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, Yes, R[x]_carrier_infinite_when_zero_ne_one)
Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, Yes, R[x]_carrier_infinite_when_zero_ne_one).
Reviewed step-by-step derivation
Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate
Canonical question JSONL · English JSONL · Validation report