Gale duality bounds for roots of polynomials with nonnegative coefficients

TL;DR

This paper introduces a novel Gale duality approach to bound the roots of polynomials with nonnegative coefficients in any basis, improving understanding of root locations and explaining clustering phenomena.

Contribution

It develops a unified Gale duality framework to analyze roots of such polynomials, incorporating linear constraints and applying to Ehrhart and chromatic polynomials.

Findings

Derived bounds for roots of polynomials with nonnegative coefficients

Applied bounds to Ehrhart and chromatic polynomials

Provided explanation for root clustering in random polynomials

Abstract

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most $d$. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.