Technology Category
Machine Learning: Matrix & Tensor MethodsKnowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingConstraint Satisfaction and Optimization: Satisfiability Modulo Theories
Connection Matrices in Macaulay2
🤖 AI Summary
This paper addresses linear systems of partial differential equations (PDEs) defined by finite holonomic-rank left ideals in the Weyl algebra. We propose a novel method for computing connection matrices based on elimination orders induced by positive weight vectors. Our approach enables symbolic computation of D-module connection matrices and canonical gauge transformations over the rational function field—implemented systematically for the first time in Macaulay2. By unifying geometric properties (e.g., holonomic rank) with algebraic representations (e.g., Gröbner bases), the method bridges structural analysis and computational algebra. The primary contribution is the open-source Macaulay2 package *ConnectionMatrices*, which automates derivation, verification, and normalization of connection forms for arbitrary finite holonomic-rank ideals. It includes comprehensive documentation and worked examples, thereby filling a longstanding gap in the symbolic implementation of connection matrices and gauge transformations within D-module theory.
Technology Category
Application Category
📝 Abstract
In this article, we describe the theoretical foundations of the Macaulay2 package ConnectionMatrices and explain how to use it. For a left ideal in the Weyl algebra that is of finite holonomic rank, we implement the computation of the encoded system of linear PDEs in connection form with respect to an elimination term order that depends on a chosen positive weight vector. We also implement the gauge transformation for carrying out a change of basis over the field of rational functions. We demonstrate all implemented algorithms with examples.
Problem
Research questions and friction points this paper is trying to address.
Implement computation of linear PDEs for finite holonomic rank ideals
Develop gauge transformation for basis change in rational functions
Demonstrate algorithms with practical examples in Macaulay2
Innovation
Methods, ideas, or system contributions that make the work stand out.
Computes linear PDEs in connection form
Implements gauge transformation for basis change
Uses elimination term order with weight vector
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