A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal

2025/04/04 by Rachel N. Diethorn, Sema Güntürkün, Diethorn, Rachel +11 · 1 citation
Computer Science · Mathematics · #13C13 #13D02 #13P10 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2504.03977

openalex publication_date 2025/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We examine the ideal I=(x₁², …, x_n², (x₁+…+x_n)²) in the polynomial ring Q=k[x₁, …, x_n], where k is a field of characteristic zero or greater than n. We also study the Gorenstein ideal G linked to I via the complete intersection ideal (x₁², …, x_n²). We compute the Betti numbers of I and G over Q when n is odd and extend known computations when n is even. A consequence is that the socle of Q/I is generated in a single degree (thus Q/I is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal G.

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• The Gröbner basis for powers of a general linear form in a monomial complete intersection

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