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

  • Alexander Dvorsky
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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, and C a polynomial with all the coefficients 0 and 1, must already be a factorization into $0−−1$ polynomials. Let k be odd and 0<a<1. We study the possibility that 1+a x2+xk divides a $0−−1$ polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case k=5. We prove that no such factorization exists for any odd k≥ 341. The intermediate cases (5 < k < 341) are treated in the companion paper.

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Cite

BibTeXRIS
BibTeX
@misc{dvorsky2026unfair,
  title = {{The Unfair 0-1 Polynomial Problem and High-Degree Trinomials}},
  author = {Alexander Dvorsky},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.19173},
  eprint = {2608.19173},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.19173},
}
RIS
TY  - UNPB
TI  - The Unfair 0-1 Polynomial Problem and High-Degree Trinomials
AU  - Alexander Dvorsky
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2608.19173
UR  - https://arxiv.org/abs/2608.19173
AB  - 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, and C a polynomial with all the coefficients 0 and 1, 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. We prove that no such factorization exists for any odd k≥ 341. The intermediate cases (5 < k < 341) are treated in the companion paper.
ER  -