On the p-irreducibility of quintic positive polynomials.

This article focuses on the p-reducibility and p-irreducibility of positive polynomials, particularly quintic (degree 5) polynomials, and explores sufficient conditions for their factorization into positive polynomial factors. Positive polynomials, defined as those with positive leading and constant coefficients and non-negative intermediate coefficients, are significant in both pure mathematics and biochemical applications, especially in modeling cooperativity in ligand–protein binding via binding polynomials. The paper distinguishes between Hurwitzian positive polynomials—whose zeros lie strictly in the left half-plane and are always p-reducible—and non-Hurwitzian polynomials, for which p-reducibility is more complex. It presents criteria and theorems (including those by Lipatov–Sokolov, Katkova–Vishnyakova, Anderson–Saff–Varga, and Xie) to test Hurwitzianity and p-(ir)reducibility, applies these to biological binding models such as the KNF model and DNA binding scenarios, and confirms cases of both p-reducible and p-irreducible quintic binding polynomials. While a complete characterization of p-irreducibility for quintic positive polynomials remains open, the methods developed provide practical tools for analyzing polynomial factorization relevant to biochemical cooperativity.