On the Log-Concavity of Hilbert Series of Veronese Subrings and Ehrhart Series

TL;DR

This paper proves that for large enough n, certain polynomial transformations related to Hilbert series and Ehrhart series have roots and coefficients with strong log-concavity and unimodality properties, with applications in algebra and combinatorics.

Contribution

It establishes the existence of a threshold n_d ensuring log-concavity and real-rootedness of transformed polynomials, advancing understanding of Hilbert and Ehrhart series.

Findings

For large n, the transformed polynomials have real, negative roots.

Coefficients of transformed polynomials are strictly log-concave and unimodal.

Applications include Ehrhart δ-polynomials and Hilbert series of Veronese subrings.

Abstract

For every positive integer $n$, consider the linear operator $\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\frac{h(t)}{(1 - t)^{d + 1}} = \sum_{m \geq 0} g(m) t^{m}$, for some polynomial $g(m)$ with rational coefficients, then $\frac{\U_{n}h(t)}{(1- t)^{d + 1}} = \sum_{m \geq 0} g(nm) t^{m}$. We show that there exists a positive integer $n_{d}$, depending only on $d$, such that if $h(t)$ is a polynomial of degree at most $d$ with nonnegative integer coefficients and $h(0) \geq 1$, then for $n \geq n_{d}$, $\U_{n}h(t)$ has simple, real, strictly negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart $\delta$-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen--MacCauley graded rings.