Totally nonnegative matrices, chain enumeration and zeros of polynomials

TL;DR

This paper establishes a connection between totally nonnegative matrices and polynomials with real zeros, developing a unified theory for chain enumeration in posets and zeros of chain polynomials, with broad mathematical implications.

Contribution

It introduces a general framework linking totally nonnegative matrices to real-rooted polynomials and extends the theory of $h$-vectors for posets, unifying several prior results.

Findings

Any lower unitriangular totally nonnegative matrix yields polynomials with only real zeros.

Characterization of the convex hull of characteristic polynomials of hyperplane arrangements.

Resolution of an open problem on the real-rootedness of polynomials from bivariate rational functions.

Abstract

We prove that any lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This has consequences for problems in several areas of mathematics. We use it to develop a general theory for chain enumeration in posets and zeros of chain polynomials. The results obtained extend and unify results of the first author, Brenti, Welker and Athanasiadis. In the process we define a notion of $h$-vectors for a large class of posets which generalize the notions of $h$-vectors associated to simplicial and cubical complexes. A consequence of our methods is a characterization of the convex hull of all characteristic polynomials of hyperplane arrangements of fixed dimension and over a fixed finite field. This may be seen as a refinement of the Critical Problem of Crapo and Rota. We also use the methods developed to answer an open problem posed by Forg\'acs and Tran on the real-rootedness of polynomials arising from certain bivariate rational functions.