Test ideals in diagonal hypersurface rings II

2002/07/12 by Moira A. McDermott, McDermott, Moira A.
Computer Science · Mathematics · #13A35 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #msc:13A35

paper · pdf · doi:10.48550/arxiv.math/0207109

revised version, incorporating referee's comments, LaTeX, 10 pages

openalex publication_date 2002/07/12 · arxiv created 2002/12/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R=k[x₁, ..., x_n]/(x₁^d + ... + x_n^d), where k is a field of characteristic p, p does not divide d and n ≥ 3. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to d) including several that are not integrally closed. These examples provide a negative answer to Smith's (2000, Comm. in Alg.) question of whether the test id eal in general is always integrally closed.

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