The Poincar\'e series of a local Gorenstein ring of multiplicity up to 10 is rational

TL;DR

This paper proves that the Poincaré series of certain local Gorenstein rings, especially those with multiplicity up to 10, are rational, extending known results to a broader class of rings.

Contribution

It introduces a specific class of local Gorenstein rings and demonstrates the rationality of their Poincaré series, including all rings with multiplicity up to 10.

Findings

Proves rationality of Poincaré series for a class of Gorenstein rings

Extends rationality results to all Gorenstein rings with multiplicity ≤ 10

Provides explicit computations of Poincaré series in the studied class

Abstract

Let $R$ be a local, Gorenstein ring with algebraically closed residue field $k$ of characteristic 0 and let $P_R(z):=\sum_{p=0}^{\infty}\dim_k(\tor_p^R(k,k))z^p$ be its Poincar\'e series. We compute $P_R$ when $R$ belongs to a particular class defined in the introduction, proving its rationality. As a by--product we prove the rationality of $P_R$ for all local, Gorenstein rings of multiplicity at most 10.