# Lorentzian Polynomials II: Applications

In this previous post, I described the basic theory of Lorentzian polynomials d’après Brändén and Huh. Now I’d like to describe some of the powerful applications of this theory, culling together results from papers by several different sets of authors. Our first application will be Mason’s Ultra-Log-Concavity Conjecture from 1972, established independently by Brändén-Huh and Anari-Liu-Oveis Gharan-Vinzant in 2018.

Theorem: Let $M$ be a matroid on $n$ elements, and let $I_k(M)$ denote the number of independent sets of size $k$ in $M$. Then the sequence $I_k(M)$ is ultra-log-concave.

A special case of this result (which seems to be no easier to prove than the general case) is the following: Let $E$ be a set of $n$ vectors in some finite-dimensional vector space, and let $I_k$ denote the number of $k$-element linearly independent subsets of $E$. Then the sequence $I_k$ is ULC.

One major goal of the theory of Lorentzian polynomials is to provide new techniques for proving that various naturally-occurring sequences of non-negative real numbers $a_k$ are log-concave, meaning that $a_k^2 \geq a_{k-1} a_{k+1}$ for all $k$. More generally, the authors consider homogeneous multivariate polynomials $p(x_1,\ldots,x_n)$ with non-negative coefficients and study certain natural extensions of log-concavity to this setting. (For some background on log-concave sequences, see this survey paper which I wrote for the Bulletin of the AMS.) So let me begin by introducing two “classical” methods for proving (an even sharper version of) log-concavity of the coefficients of certain polynomials.