A family of local rings with rational Poincaré series

Abstract

Abstract. In this note we compute the Poincaré series of almost stretched Gorenstein local rings. It turns out that it is rational.

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

“…The above theorems generalize the quoted results on stretched, almost-stretched and short algebras (see [16], [10], [6], [8]).…”

“…The rationality of the Poincaré series P A of every stretched ring A is proved in [16]. The proof has been generalized to rings with H A (2) = 2 in [10] and to rings with H A (2) = 3, H A (3) = 1 in [6] . The rationality of P A when A is a 2-stretched algebra has been studied in [8] with the restriction sdeg(A) = 3.…”

“…We now focus on the Poincaré series P A (z) of the algebra A defined in the introduction: we will generalize some classical results (see [16], [10], [6]). Out of the decomposition results proved in the previous section, the main tools we use are the following ones:…”

“…The rationality of the Poincaré series of stretched algebras is proved in [16]. For almost stretched algebras see [10]. For the case of algebras A with sdeg(A) = 3 and H A (2) ≤ 4 see [8].…”

Poincaré Series and Deformations of Gorenstein Local Algebras

“…Remark 3.5. J. Elias and G. Valla proved that if the multiplicity of A is either d ≤ 9 or d ≤ h + 4, then P A (z) is rational (see [18]). If d ≤ h + 6 then by Corollary 3.2 and [18] we get that P A (z) is rational but the case that A has Hilbert function (1, h, 3, 1, 1) is still open.…”