Chow polynomials of uniform matroids are real-rooted

TL;DR

This paper proves that the Chow polynomials of uniform matroids and certain posets have only real roots, confirming a conjecture for these classes and advancing understanding of their algebraic properties.

Contribution

The paper establishes the real-rootedness of Chow polynomials for uniform matroids and maximally ranked posets, confirming a conjecture for these cases.

Findings

Chow polynomials of uniform matroids are real-rooted.

Chow and augmented Chow polynomials of maximally ranked posets are real-rooted.

Supports the conjecture that Hilbert-Poincaré series of matroid Chow rings have only real zeros.

Abstract

June Huh and Matthew Stevens conjectured that the Hilbert-Poincar\'e series of the Chow ring of any matroid is a polynomial with only real zeros. We prove this conjecture for the class of uniform matroids. We also prove that the Chow polynomial and the augmented Chow polynomial of any maximally ranked poset has only real zeros.