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2016 On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence
Gianfranco Casnati, Joachim Jelisiejew, Roberto Notari
Rocky Mountain J. Math. 46(2): 413-433 (2016). DOI: 10.1216/RMJ-2016-46-2-413

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

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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

Information

Published: 2016
First available in Project Euclid: 26 July 2016

zbMATH: 06624467
MathSciNet: MR3529076
Digital Object Identifier: 10.1216/RMJ-2016-46-2-413

Subjects:
Primary: 13D40
Secondary: 13H10

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

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Vol.46 • No. 2 • 2016
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