An Algorithm to Construct Groebner Bases for Solving Integration by Parts Relations
A. V. Smirnov
TL;DR
Addresses reduction of large families of Feynman integrals F(a_1,...,a_n) by exploiting IBP relations with a generalized Buchberger algorithm for shift-operator algebras. Develops the s-reduction framework and constructs sector-specific s-bases to produce finite sets of master integrals while preserving boundary conditions. Introduces s-form, nu-degrees, and s-polynomials to guide a Buchberger-like reduction; demonstrates substantial efficiency gains in two- and three-loop examples and suggests applicability up to n≈12 indices, with Mathematica implementation available. Overall, the work presents a practical, scalable alternative to existing IBP-reduction methods for perturbative calculations in quantum field theory.
Abstract
This paper is a detailed description of an algorithm based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. The algorithm is used for calculating Feynman integrals and has proven itself efficient in several complicated cases.