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Question 9399: Nonnegative Polynomial Degrees Including Both Zero Polynomials

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0017-09399 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0049

Original question and givens

Fix the cost model. Coefficients in ascending powers are f=[23, 3, 1], g=[1, 24, 1, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.

  • G001 — Fix the cost model. Coefficients in ascending powers are f=[23, 3, 1], g=[1, 24, 1, 1]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

fixed cost model;explicit nonnegative functions on natural n

After

O,Omega and Theta quantify positive constants over a full tail

Conditions: none

Fix the cost model and function family first.O requires C>0 and N with an upper bound for every n≥N;Omega is the corresponding positive lower bound,and Theta requires both.Finite measurements describe only a finite window.

Step identifiers and dependencies

Rule: full_tail_not_finite_benchmark_scope · Uses: G001

S002 · calculation

Before

f coefficient sum := sum(23, 3, 1)

After

f coefficient sum = 27

Conditions: none

Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

polynomial=f;original ascending coefficients=[23, 3, 1]

After

nonzero indices=[0, 1, 2];degree=2;coefficient sum=27;leading coefficient/status=1

Conditions: none

Retain the original coefficient vector including trailing zeros.The highest nonzero index is the degree;the zero polynomial has degree−1.For a nonzero polynomial with nonnegative coefficients and n≥1,leading*n^degree≤P(n)≤sum(coeff)*n^degree,so no leading cancellation occurs.

Step identifiers and dependencies

Rule: nonnegative_polynomial_degree_and_bounds · Uses: G001, S002

S004 · calculation

Before

g coefficient sum := sum(1, 24, 1, 1)

After

g coefficient sum = 27

Conditions: none

Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

polynomial=g;original ascending coefficients=[1, 24, 1, 1]

After

nonzero indices=[0, 1, 2, 3];degree=3;coefficient sum=27;leading coefficient/status=1

Conditions: none

Retain the original coefficient vector including trailing zeros.The highest nonzero index is the degree;the zero polynomial has degree−1.For a nonzero polynomial with nonnegative coefficients and n≥1,leading*n^degree≤P(n)≤sum(coeff)*n^degree,so no leading cancellation occurs.

Step identifiers and dependencies

Rule: nonnegative_polynomial_degree_and_bounds · Uses: G001, S004

S006 · implies

Before

direction=f versus g;degrees=(2,3);sums=(27,27);leading=(1,1)

After

Big-O=Yes;argument=C=27,N=1;upper numerator <= 27*n^2 <= C*1*n^3 <= C*denominator

Conditions: none

Separate zero and nonzero-degree cases.A success supplies positive C and a full-tail bound;a failure gives a larger integer n for every proposed C,N,not just a finite sample.Omega is obtained by swapping numerator/denominator in the equivalent O condition and reciprocating a positive constant.

Step identifiers and dependencies

Rule: polynomial_quantified_upper_bound_case · Uses: G001, S005

S007 · implies

Before

direction=g versus f (Omega);degrees=(3,2);sums=(27,27);leading=(1,1)

After

Big-O=No;argument=for any C>0,N choose integer n>=max(N,1) and n>C*27/1;positive exponent gap=1 forces numerator>C*denominator

Conditions: none

Separate zero and nonzero-degree cases.A success supplies positive C and a full-tail bound;a failure gives a larger integer n for every proposed C,N,not just a finite sample.Omega is obtained by swapping numerator/denominator in the equivalent O condition and reciprocating a positive constant.

Step identifiers and dependencies

Rule: polynomial_quantified_upper_bound_case · Uses: G001, S006

S008 · implies

Before

O(f,g)=Yes;O(g,f)=No

After

O=Yes;Omega=No;Theta=No

Conditions: none

For zero versus zero both bounds0≤C*0 hold.Zero versus nonzero gives only the upper class;nonzero versus zero gives only the lower class.Compare degrees only in nonzero cases;degree−1 is not a nonzero constant degree.

Step identifiers and dependencies

Rule: zero_polynomial_tail_class_separation · Uses: G001, S007

Reverse checks

  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Recompute g coefficient sum with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Recompute f coefficient sum with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([23, 3, 1], [1, 24, 1, 1], 2, 3, Yes, No, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: de8534964b0aeb6749f2bb2ffb0ecdff7d2b17a255ea0d7790c18b15ec8af56d
  • candidate record SHA-256: 7d21ecf2579108058b9d4938f10b0d390890ca12d925272f53b7251d3342684b
  • semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
  • question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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