Homological Methods for Hypergeometric Families

2004/06/18 by Laura Felicia Matusevich, Ezra Miller, Matusevich, Laura Felicia +3 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG

paper · pdf · doi:10.48550/arxiv.math/0406383

openalex publication_date 2004/06/18 · arxiv created 2004/06/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems H_A(β) arising from a d x n integer matrix A and a parameter β∈ \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^d, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter βis rank-jumping for H_A(β) if and only if βlies in the Zariski closure of the set of \ZZ^d-graded degrees αwhere the local cohomology \bigoplus_i

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• Characterizing Cohen-Macaulay One-Loop Feynman Integrals
• Hypergeometric Discriminants

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