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Comprehensive Restriction Algorithm for Hypergeometric Systems

Hiromasa Nakayama, Nobuki Takayama

TL;DR

This work develops a comprehensive framework for restricting holonomic $D$-modules with parameters to linear subspaces, extending Oaku's restriction algorithm to parameterized families via comprehensive Gröbner systems and generic $b$-functions. It integrates contiguity-relations methods, isomorphism constructions for $A$-hypergeometric and Horn-type systems, and a comprehensive restriction algorithm that stratifies parameter spaces and yields explicit restricted modules, including Gauss and Appell hypergeometric systems. The key contributions include algorithms for deriving contiguity operators under parameter restrictions, transferring contiguity from $A$-hypergeometric to classical hypergeometric systems, and providing concrete restriction results for Gauss and Appell $F_1$ systems. The results enhance our ability to compute parameter-stratified, parameter-independent restrictions and to characterize isomorphism classes, with potential impact on symbolic computation and the study of hypergeometric $D$-modules in mathematical physics and algebraic geometry.

Abstract

An algorithm computing the restriction of a holonomic D-module to a linear subspace was given by T.Oaku in 1997. We consider a problem of computing the restriction for a given holonomic D-module with parameters. We will give a partial answer to the problem for general holonomic D-modules and an answer to hypergeometric holonomic D-modules.

Comprehensive Restriction Algorithm for Hypergeometric Systems

TL;DR

This work develops a comprehensive framework for restricting holonomic -modules with parameters to linear subspaces, extending Oaku's restriction algorithm to parameterized families via comprehensive Gröbner systems and generic -functions. It integrates contiguity-relations methods, isomorphism constructions for -hypergeometric and Horn-type systems, and a comprehensive restriction algorithm that stratifies parameter spaces and yields explicit restricted modules, including Gauss and Appell hypergeometric systems. The key contributions include algorithms for deriving contiguity operators under parameter restrictions, transferring contiguity from -hypergeometric to classical hypergeometric systems, and providing concrete restriction results for Gauss and Appell systems. The results enhance our ability to compute parameter-stratified, parameter-independent restrictions and to characterize isomorphism classes, with potential impact on symbolic computation and the study of hypergeometric -modules in mathematical physics and algebraic geometry.

Abstract

An algorithm computing the restriction of a holonomic D-module to a linear subspace was given by T.Oaku in 1997. We consider a problem of computing the restriction for a given holonomic D-module with parameters. We will give a partial answer to the problem for general holonomic D-modules and an answer to hypergeometric holonomic D-modules.

Paper Structure

This paper contains 14 sections, 7 theorems, 114 equations, 1 figure, 7 algorithms.

Table of Contents

  1. Introduction
  2. Comprehensive Gröbner System and Generic $b$-function
  3. Review on Algorithms for Contiguity Relations
  4. Deriving down(up)-step operator for a given up(down)-step contiguity operator
  5. Finding contiguity operator for rigid systems
  6. Finding isomorphisms among $A$-hypergeometric systems
  7. Finding contiguity operators by finding rational solutions
  8. From contiguity operators of $A$-hypergeometric systems to those of classical hypergeometric systems
  9. Restriction of a left $D$-homomorphism
  10. Restriction of isomorphisms of $A$-hypergeometric systems to those of classical hypergeometric systems --- restriction of Saito's isomorphism
  11. Representatives of isomorphic classes
  12. Comprehensive Restriction Algorithm
  13. Restriction of the Gauss Hypergeometric System to the Origin
  14. Restriction of Appell $F_1$ System to the Origin

Key Result

Theorem 1

Assume $A=\left( E_d \ \vert \ A' \right)$.

Figures (1)

  • Figure 1: A part of the directed graph and reducing representatives

Theorems & Definitions (30)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 20 more