Finding Linear Dependencies in Integration-By-Parts Equations: A Monte Carlo Approach
Philipp Kant
TL;DR
The paper tackles the bottleneck in multiloop IBP reductions caused by a large, redundancy-rich set of IBP/LI equations. It introduces a Monte Carlo finite-field algorithm that maps polynomial-coefficient systems to $\mathbb{F}_p$ and uses Gaussian elimination to identify a maximal linearly independent subset, reducing both memory and CPU time for Laporta reductions. The method also yields the master integrals and can be tuned to optimize equation selection; tests on vacuum topologies show substantial removal of redundant equations with results agreeing with known literature. The approach is efficient, scales with the number of kinematic invariants, and can be augmented with orthogonal checks to yield a Las Vegas-style verification.
Abstract
The reduction of a large number of scalar integrals to a small set of master integrals via Laporta's algorithm is common practice in multi-loop calculations. It is also a major bottleneck in terms of running time and memory consumption. It involves solving a large set of linear equations where many of the equations are linearly dependent. We propose a simple algorithm that eliminates all linearly dependent equations from a given system, reducing the time and space requirements of a subsequent run of Laporta's algorithm.