Dowling's polynomial conjecture for independent sets of matroids

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.