Symmetric decompositions and the Veronese construction

Abstract: We study rational generating functions of sequences ${a_n}{n\geq 0}$ that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences ${a{rn}}{n\geq 0}$. We prove that if the numerator polynomial for ${a_n}{n\geq 0}$ is of degree $s$ and its coefficients satisfy a set of natural linear inequalities then the symmetric decomposition of the numerator for ${a_{rn}}{n\geq 0}$ is real-rooted whenever $r\geq \max {s,d+1-s}$. Moreover, if the numerator polynomial for ${a_n}{n\geq 0}$ is symmetric then we show that the symmetric decomposition for ${a_{rn}}_{n\geq 0}$ is interlacing. We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the $h^\ast$-polynomial of every dilation of a $d$-dimensional lattice polytope of degree $s$ has a real-rooted symmetric decomposition whenever the dilation factor $r$ satisfies $r\geq \max {s,d+1-s}$. Moreover, if the polytope is Gorenstein then this decomposition is interlacing.