# 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.

# Infinitely many primes via generating functions

A few years ago I discovered an amusing proof of Euclid’s theorem that there are infinitely many primes which I thought I’d record here for posterity. (I subsequently learned that a similar argument appears in this paper by Paul Pollack.)

To motivate the proof, and illustrate its working in an (admittedly silly) special case, suppose that there were just two prime numbers, called p and q. Then by the fundamental theorem of arithmetic (i.e., unique factorization into primes) we would have the following identity between generating functions:

$\sum_{n=2}^\infty z^n = \sum_{k=1}^\infty z^{kp} + \sum_{k=1}^\infty z^{kq} - \sum_{k=1}^\infty z^{kpq}.$

Indeed, there is precisely one term $z^n$ for each integer $n \geq 2$ on the left-hand side, and the same is true for the right-hand side (consider separately the cases where $p \mid n$ but $q \nmid n$, $q \mid n$ but $p \nmid n$, and $pq \mid n$). Using the formula for the sum of a geometric series, we can rewrite our formula as an identity between complex-analytic functions valid on the open unit disc ${\mathbb D} = \{z \in {\mathbb C} \; : \; |z|<1 \}$:

$\frac{z^2}{1-z} = \frac{z^p}{1-z^p} + \frac{z^q}{1-z^q} - \frac{z^{pq}}{1-z^{pq}}.$

This is impossible, however, as we see by letting $z$ approach a primitive $pq^{\rm th}$ root of unity, since each term stays bounded except for $\frac{z^{pq}}{1-z^{pq}}$, which tends to infinity.

A drunken man standing one step from the edge of a cliff takes a sequence of random steps either away from or toward the cliff. At any step, his probability of taking a step away from the cliff is $p$ (and therefore his probability of taking a step toward the cliff is $1-p$). What is the probability that the man will eventually fall off the cliff?