# Question 9393: Nonnegative Polynomial Degrees Including Both Zero Polynomials [Return to catalogue page 47](https://ocme.evemisslab.com/en/questions/page/47/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations/#q-20261013-advance-20261019-r06-asymptotic-resource-bound-0043) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0017-09393 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0043 ## Original question and givens Fix the cost model. Coefficients in ascending powers are f=[17, 3, 1], g=[1, 18, 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=[17, 3, 1], g=[1, 18, 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. - [mko-asymptotic-resource-bound](https://ocme.evemisslab.com/en/mko/mko-asymptotic-resource-bound/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: full_tail_not_finite_benchmark_scope · Uses: G001 S002 · calculationBefore f coefficient sum := sum(17, 3, 1) After f coefficient sum = 21 Conditions: none Calculate exactly in the specified source-operand order without approximation or substituting a different candidate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore polynomial=f;original ascending coefficients=[17, 3, 1] After nonzero indices=[0, 1, 2];degree=2;coefficient sum=21;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 dependenciesRule: nonnegative_polynomial_degree_and_bounds · Uses: G001, S002 S004 · calculationBefore g coefficient sum := sum(1, 18, 1, 1) After g coefficient sum = 21 Conditions: none Calculate exactly in the specified source-operand order without approximation or substituting a different candidate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore polynomial=g;original ascending coefficients=[1, 18, 1, 1] After nonzero indices=[0, 1, 2, 3];degree=3;coefficient sum=21;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 dependenciesRule: nonnegative_polynomial_degree_and_bounds · Uses: G001, S004 S006 · impliesBefore direction=f versus g;degrees=(2,3);sums=(21,21);leading=(1,1) After Big-O=Yes;argument=C=21,N=1;upper numerator <= 21*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 dependenciesRule: polynomial_quantified_upper_bound_case · Uses: G001, S005 S007 · impliesBefore direction=g versus f (Omega);degrees=(3,2);sums=(21,21);leading=(1,1) After Big-O=No;argument=for any C>0,N choose integer n>=max(N,1) and n>C*21/1;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 dependenciesRule: polynomial_quantified_upper_bound_case · Uses: G001, S006 S008 · impliesBefore 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 dependenciesRule: 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 ([17, 3, 1], [1, 18, 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: 4a4aa8a342a61a800a285e1458ffc79161612b8f4a7964b21a4bfa704997331b - candidate record SHA-256: 59520df665da89da79ee7aed3ce67df533e0a35cfe21037cca2b1fcb38e1932d - semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8 - question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r06-asymptotic-resource-bound-0043.json)