The Unfair 0-1 Polynomial Problem and High-Degree Trinomials · arXivDesk2608.19173Aug 19, 2026
The Unfair 0-1 Polynomial Problem and High-Degree Trinomials
Alexander Dvorsky
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
--
polynomials. Let
be odd and
. We study the possibility that
1+ax2+xk divides a
--
polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case
, and the cases
were treated subsequently by finite recurrence and spectral arguments. We prove that no such factorization exists for any odd
.