On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

Abstract

Let $A$ be a local Artinian Gorenstein ring with algebraically closed residue field $A/\mathfrak{M}=k$ of characteristic 0, and let $P_A(z):={\sum }_{p=0}^{\infty }({\mathrm{Tor}}_p^A(k,k))z^p$ be its Poincaré series. We prove that $P_A(z)$ is rational if either ${\dim }_k({\mathfrak{M}}^2/{\mathfrak{M}}^3)\le 4$ and ${\dim }_k(A)\le 16,$ or there exist $m\le 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $m$ for $n\in [2,c]$ and equal to 1 for $n>c$. The results are obtained thanks to a decomposition of the apolar ideal $\mathrm{Ann}(F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.

12 pages - The section on preliminary results has been revised. Several other minor corrections have been made in the whole paper

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