Abstract
Let $A$ be a local Artinian Gorenstein algebra with maximal ideal $\fM $, \[P_A(z) := \sum _{p=0}^{\infty } (\tor _p^A(k,k))z^p \] its Poicar\'{e} series. We prove that $P_A(z)$ is rational if either $\dim _k({\fM ^2/\fM ^3}) \leq 4 $ and $ \dim _k(A) \leq 16,$ or there exist $m\leq 4$ 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 $n =c+1$. The results are obtained due to a decomposition of the apolar ideal $\Ann (F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.
Citation
Gianfranco Casnati. Joachim Jelisiejew. Roberto Notari. "On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence." Rocky Mountain J. Math. 46 (2) 413 - 433, 2016. https://doi.org/10.1216/RMJ-2016-46-2-413
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