Volume Polynomials and Log-concavity of the Characteristic Polynomials of Matroids

TL;DR

This paper explains the proof of the log-concavity of matroid characteristic polynomials using Lorentzian polynomials, making the complex proof accessible to students.

Contribution

It provides an accessible exposition of the Lorentzian polynomial proof of the Rota--Heron--Welsh conjecture for undergraduates and beginners.

Findings

Confirmed the log-concavity of matroid characteristic polynomials.

Presented an accessible proof using Lorentzian polynomials.

Clarified the connection between matroid theory and Lorentzian polynomials.

Abstract

The Rota--Heron--Welsh conjecture (now a theorem of Adiprasito, Huh, and the author) asserts the log-concavity of the characteristic polynomial of matroids. We give an exposition of the Lorentzian polynomial proof following the work of Branden and Leake aimed at undergraduate and beginning graduate students.