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

For every positive integer $n$, consider the linear operator ${\text{U}}_n$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\frac{h(t)}{(1-t{)}^{d+1}}={\sum }_{m\ge 0}g(m)t^m$, 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 $h(0)\ge 1$, then for $n\ge n_d$, ${\text{U}}_{n}h(t)$ 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.