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 be a matroid on elements, and let denote the number of independent sets of size in . Then the sequence 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 be a set of vectors in some finite-dimensional vector space, and let denote the number of -element linearly independent subsets of . Then the sequence is ULC.

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