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