# Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

## TL;DR

This paper presents a novel algebro-geometric IBP reduction method for complex Feynman integrals, leveraging parallel computation frameworks to automate and optimize the reduction process.

## Contribution

It introduces a new IBP reduction approach combining algebraic geometry, sparse linear algebra, and parallel computing with Singular and GPI-Space.

## Key findings

- Successfully reduced two-loop five-point nonplanar double-pentagon integrals.
- Automated and parallelized the IBP reduction process.
- Potential for simplifying IBP coefficients in transcendental bases.

## Abstract

We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficiently automatized and automatically parallelized by modeling the algorithm in GPI-Space using the language of Petri-nets. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.