Log-concavity of asymptotic multigraded Hilbert series

TL;DR

This paper investigates the asymptotic behavior of multigraded Hilbert series, revealing log-concavity of coefficients and extending known results beyond standard grading, with implications for algebraic and combinatorial structures.

Contribution

It provides a polyhedral description of the asymptotic polynomial and proves the log-concavity of its coefficients, extending prior work to multigraded contexts.

Findings

Coefficients of the asymptotic polynomial are log-concave.

Asymptotic behavior depends on multidegree and the multigraded polynomial ring.

Extended results of Beck-Stapledon and Diaconis-Fulman beyond standard grading.

Abstract

We study the linear map sending the numerator of the rational function representing the Hilbert series of a module to that of its r-th Veronese submodule. We show that the asymptotic behaviour as r tends to infinity depends on the multidegree of the module and the underlying positively multigraded polynomial ring. More importantly, we give a polyhedral description for the asymptotic polynomial and prove that the coefficients are log-concave. In particular, we extend some results by Beck-Stapledon and Diaconis-Fulman beyond the standard graded situation.