Preservation of log-concavity on gamma polynomials

TL;DR

This paper proves that log-concavity of the gamma polynomial of a symmetric polynomial implies the log-concavity of the polynomial itself, extending previous results about root properties and ultra log-concavity.

Contribution

It establishes that log-concavity of the gamma polynomial ensures the log-concavity of the original symmetric polynomial, answering an open question.

Findings

Log-concavity of gamma polynomial implies log-concavity of the polynomial

Established a binomial inequality via path-counting argument

Extended known root and ultra log-concavity results

Abstract

Every symmetric polynomial $h(x)$ with center of symmetry $n/2$ can be expressed as a linear combination in the basis $x^i(1+x)^{n-2i}$. The $\gamma$-polynomial of $h(x)$, which we denote $\gamma_h(x)$, records the coefficients of this linear combination. Two decades ago, Br\"and\'en and Gal independently showed that if $\gamma_h(x)$ has nonpositive real roots only, then so does $h(x)$. More recently, Br\"and\'en, Ferroni, and Jochemko proved using Lorentzian polynomials that if $\gamma_h(x)$ is ultra log-concave, then so is $h(x)$, and they raised the question of whether a similar statement can be proved for the usual notion of log-concavity. The purpose of this article is to show that the answer to the question of Br\"and\'en, Ferroni, and Jochemko is affirmative. One of the crucial ingredients of the proof is an inequality involving binomial numbers that we establish via a path-counting argument.