Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture

TL;DR

This paper presents a concise proof of the log-concavity of matroid characteristic polynomial coefficients using Lorentzian polynomials extended to convex cones, and reestablishes key Hodge-Riemann relations for matroid Chow rings.

Contribution

It extends Lorentzian polynomial theory to convex cones and provides a new, simplified proof of a fundamental matroid property.

Findings

Proof of log-concavity of matroid characteristic polynomial coefficients

Extension of Lorentzian polynomials to convex cones

Reproof of Hodge-Riemann relations for matroid Chow rings

Abstract

We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convex cones, and reproves the Hodge-Riemann relations of degree one for the Chow ring of a matroid.