On the real-rootedness of the Veronese construction for rational formal power series

Abstract

We study real sequences $\{a_n{\}}_{n\in \mathbb{N}}$ that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree $s$ and has only nonnegative coefficients, then the numerator polynomial of the subsequence $\{a_{rn+i}{\}}_{n\in \mathbb{N}}$, $0\le i<r$, has only nonpositive, real roots for all $r\ge s-i$. We apply our results to combinatorially positive valuations on polytopes and to Hilbert functions of Veronese submodules of graded Cohen-Macaulay algebras. In particular, we prove that the Ehrhart $h^{\ast }$-polynomial of the $r$-th dilate of a $d$-dimensional polytope has only distinct, negative, real roots if $r\ge \min \{s+1,d\}$. This proves a conjecture of Beck and Stapledon (2010).

arXiv comment14 pages; improved bounds; section 5 and 6 on optimality added