Preservation of log-concavity on gamma polynomials

Source abstract

Every symmetric polynomial h ( x ) h(x) with center of symmetry n / 2 n/2 can be expressed as a linear combination in the basis x i ( 1 + x ) n − 2 i x^i(1+x)^{n-2i} . The γ \gamma -polynomial of h ( x ) h(x) , which we denote γ h ( x ) \gamma _h(x) , records the coefficients of this linear combination. Two decades ago, Brändén [Electron. J. Combin. 11 (2004/06), Research Paper 9] and Gal [Discrete Comput. Geom. 34 (2005), pp. 269–284] independently showed that if γ h ( x ) \gamma _h(x) has nonpositive real roots only, then so does h ( x ) h(x) . More recently, Brändén, Ferroni, and Jochemko [ Preservation of inequalities under Hadamard products , (2024) Preprint, arXiv: 2408.12386 ] proved using Lorentzian polynomials that if γ h ( x ) \gamma _h(x) is ultra log-concave, then so is h ( x ) 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 Brändén, 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.