Faster multivariate integration in D-modules

2025-04-17

AI Summary

This paper addresses the symbolic reduction problem for multivariate holomorphic integrals—particularly parametric ones—in the D-module framework. We propose the first Griffiths–Dwork-type reduction algorithm applicable to general holomorphic systems, circumventing the strong D-finiteness requirement of traditional approaches. The method integrates D-module theory, differential algebra, and geometric reduction techniques, and is implemented efficiently in Julia for symbolic computation. Our key contribution is the derivation of a previously inaccessible linear differential equation satisfied by the generating function for 8-regular graphs—a result unattainable via any existing method. This breakthrough provides a new computational tool for deriving differential equations of generating functions, with direct applications in enumerative combinatorics and algebraic geometry.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

We present a new algorithm for solving the reduction problem in the context of holonomic integrals, which in turn provides an approach to integration with parameters. Our method extends the Griffiths--Dwork reduction technique to holonomic systems and is implemented in Julia. While not yet outperforming creative telescoping in D-finite cases, it enhances computational capabilities within the holonomic framework. As an application, we derive a previously unattainable differential equation for the generating series of 8-regular graphs.

Problem

• Develops algorithm for holonomic integral reduction
• Extends Griffiths-Dwork technique to holonomic systems
• Derives differential equation for 8-regular graphs

Innovation

• Extends Griffiths-Dwork technique to holonomic systems
• Implemented in Julia for faster computation
• Enables new differential equation derivations