Tag Archives: Lorentzian polynomials

The geometry of snowflaked Ptolemaic metric spaces

My coauthors June Huh, Mario Kummer, Oliver Lorscheid, and I just posted a paper called Lorentzian polynomials and matroids over triangular hyperfields, Part 2: Analytic aspects on the arXiv preprint server. This is the first paper of mine to contain a substantial result proved by AI, and also the first to contain a proof formally verified in Lean. So I thought I would take this opportunity to tell a bit of the backstory.

Although our paper is motivated by the general theory of Lorentzian polynomials and matroids with coefficients, which are a bit daunting to explain in a short post, the parts in which AI played a decisive role can be explained in a completely elementary way. I will first state some of our main results, as well as our main open conjecture, in terms of basic linear algebra. Then I will discuss how AI—primarily Chat GPT-5.5—was able to prove a theorem that had previously eluded us. Continue reading →

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.

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Lorentzian polynomials I: Theory

I’m organizing a reading seminar this semester on Lorentzian polynomials, mainly following this paper by Brändén and Huh but also covering some of the work of Anari et. al. In this post, I’d like to give a quick introduction to this active and beautiful subject. I’ll concentrate on the basic theory for now, and in a follow-up post I’ll discuss some of the striking applications of this theory.

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.

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