The Poincaré series of a local Gorenstein ring of multiplicity up to 10 is rational

Casnati, Gianfranco; Notari, Roberto

The Poincaré series of a local Gorenstein ring of multiplicity up to 10 is rational

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Description

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 Poincaré 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.