$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

TL;DR

This paper compares $D$-module methods with traditional approaches for solving differential equations related to Feynman integrals, demonstrating the effectiveness of algebraic techniques in deriving solutions.

Contribution

It introduces a systematic comparison between $D$-module techniques and existing methods for Feynman integral differential equations, including implementation of an algorithm for canonical series solutions.

Findings

$D$-module methods effectively derive canonical series solutions.

Comparison shows advantages of algebraic methods over traditional asymptotic series.

Provides a conceptual dictionary linking $D$-modules and Feynman integral solutions.

Abstract

Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare $D$-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic $D$-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.