The Unfair 0-1 Polynomial Problem and High-Degree Trinomials

Abstract

The unfair $0$--$1$ polynomial conjecture asks whether a factorization \[C(x)=A(x)B(x),\] with $A$ and $B$ monic and having nonnegative real coefficients, must already be a factorization into $0$--$1$ polynomials. Let $k$ be odd and $0<a<1$. We study the possibility that \[1+a x^2+x^k\] divides a $0$--$1$ polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case $k=5$, and the cases $k=7,9,11$ were treated subsequently by finite recurrence and spectral arguments. We prove that no such factorization exists for any odd $k\ge 341$.

Disclosure

“the common pre-zero interior profile, after which Sections 9 and 10 finish the proof. All genuinely numerical material is collected in the appendices. Use of AI tools. During the preparation of this paper, the author used OpenAI ChatGPT (GPT-5.6 Sol) as an interactive research assistant. Its roles included exploratory numerical work, discovering and checking quantitative estimates, generating verification code, checking the consistency of the exposition with the numerical evidence”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-22T18:17:30.791983Z

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