Faster multivariate integration in D-modules
Hadrien Brochet, Frédéric Chyzak, Pierre Lairez
TL;DR
This work advances symbolic integration for holonomic systems by introducing a reduction-based approach that extends Griffiths–Dwork techniques to holonomic $W_{t,\mathbf{x}}$-modules and enables integration with respect to parameters. The core idea is to compute normal forms modulo $\bm{\partial}M$ via a two-stage reduction that leverages left-Gröbner-basis reductions and right-derivative reductions, producing a finite-dimensional quotient when holonomic. A practical Julia implementation (MultivariateCreativeTelescoping.jl) combines memoization, confinement, and modular evaluation/interpolation to derive telescopers and differential equations, demonstrated on the generating series of $8$-regular graphs. The results show clear gains in expressivity and scalability within holonomic frameworks, with competitive performance and new differential equations that previous D-finite methods could not readily obtain. Overall, the paper broadens the toolkit for parametric integration in algebraic combinatorics and mathematical physics, offering a concrete path to new ODEs for complex generating functions.
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.