On the rationality of Poincar 'e series of Gorenstein algebras via\n Macaulay's correspondence

2013/07/05 by Gianfranco Casnati, Casnati, Gianfranco, Joachim Jelisiejew +3
Computer Science · Mathematics · #13D40 #13H10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1307.1676

openalex publication_date 2013/07/05 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let A be a local Artinian Gorenstein ring with algebraically closed residue\nfield A/ frak M=k of characteristic 0, and let P_A(z) :=\n\∑_p=0^\∞ ( Tor_p^A(k,k))z^p be its Poincar 'e series.\nWe prove that P_A(z) is rational if either \dim_k( frak M²/ frak M³)\n\≤ 4 and \dim_k(A) \≤ 16, or there exist m\≤ 4 and c such that\nthe Hilbert function H_A(n) of A is equal to m for n\∈ [2,c] and\nequal to 1 for n > c. The results are obtained thanks to a decomposition of\nthe apolar ideal mathrm Ann(F) when F=G+H and G and H belong to\npolynomial rings in different variables.\n

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