Polynomials that preserve nonnegative matrices

TL;DR

This paper investigates the class of polynomials that preserve nonnegative matrices, providing new insights into their coefficient structures and generalizations, which contributes to understanding the nonnegative inverse eigenvalue problem.

Contribution

It characterizes polynomials in _n with specific coefficient properties and introduces generalizations for even and odd parts, linking to existing literature.

Findings

Polynomials with negative coefficients can preserve nonnegative matrices.

Generalizations for even and odd parts are equivalent to known constructions.

Implications for the nonnegative inverse eigenvalue problem are discussed.

Abstract

In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83--90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this work, novel results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed.