Inequalities for Chow Polynomials and Chern Numbers of Matroids

TL;DR

This paper establishes new inequalities for Chow polynomials and Chern numbers of matroids by interpreting Chow coefficients probabilistically and connecting to algebraic geometry, leading to bounds on flats and roots.

Contribution

It introduces moment inequalities for Chow coefficients, relates them to algebraic geometry, and derives new bounds and inequalities for matroid invariants.

Findings

Bounds on the number of flags of flats

Inequalities on roots of the Chow polynomial

Chern number inequality c_1 c_{d-1} ≤ c_d

Abstract

The Chow polynomial of a matroid is a fundamental invariant whose coefficients exhibit strong positivity properties, including $\gamma$-positivity. We interpret the normalized Chow coefficients as a probability distribution and establish new inequalities for its central moments. As consequences, we obtain bounds on the number of flags of flats and inequalities on the roots of the Chow polynomial. We further relate these moment inequalities to algebraic geometry via the Hirzebruch $\chi_y$-genus. This yields new inequalities for matroidal Chern numbers. In particular, for any matroid of rank $d+1$, we prove that $c_1c_{d-1}\le c_d$, with equality if and only if $d=1$ or the simplification of the matroid is Boolean.