Dowling's polynomial conjecture for independent sets of matroids

TL;DR

This paper proves Dowling's polynomial conjecture for matroids by applying Lorentzian polynomial theory, advancing the understanding of polynomial analogues of independent set sequences.

Contribution

It provides a complete proof of Dowling's polynomial conjecture, connecting Lorentzian polynomials with matroid theory.

Findings

Dowling's polynomial conjecture is fully proven.

Lorentzian polynomials are effective in matroid theory.

Advances the polynomial version of Mason's conjecture.

Abstract

The celebrated Mason's conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason's conjecture was independently solved by Anari, Liu, Oveis Gharan and Vinzant, and by Br\"and\'en and Huh. The weak form of Mason's conjecture was also generalized to a polynomial version by Dowling in 1980 by considering certain polynomial analogue of independent set numbers. In this paper we completely solve Dowling's polynomial conjecture by using the theory of Lorentzian polynomials.