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

TL;DR

This paper investigates the real-rootedness of numerator polynomials of rational generating series and their subsequences, establishing conditions under which these polynomials have only negative, real roots, with applications to polytopes and algebraic structures.

Contribution

It proves that certain subsequences of rational generating series have numerator polynomials with only negative, real roots, confirming a conjecture related to Ehrhart polynomials.

Findings

Subsequences have numerator polynomials with only nonpositive, real roots.

Ehrhart $h^*$-polynomial of dilated polytopes has distinct, negative, real roots under certain conditions.

Results apply to valuations on polytopes and Hilbert functions of graded algebras.

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 $\{ a_{rn+i}\}_{n\in \mathbb{N}}$, $0\leq i<r$, has only nonpositive, real roots for all $r\geq s-i$. 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 $r\geq \min \{s+1,d\}$. This proves a conjecture of Beck and Stapledon (2010).