S-bases as a tool to solve reduction problems for Feynman integrals

TL;DR

This paper addresses the reduction problem for families of Feynman integrals via integration by parts (IBP) relations and proposes a formal definition of master integrals. It surveys algorithmic approaches, including Laporta’s linear-system method and Gröbner-bases–based strategies, and introduces the 2SAS06 method that uses s-bases with sector-aware ordering. A central contribution is a rigorous framing of master integrals through a solution space to the relations, accompanied by constructive criteria for certification. The authors illustrate the method with a three-loop example, listing explicit master integrals and reduction formulas, demonstrating practical applicability to complex multi-loop problems.

Abstract

We suggest a mathematical definition of the notion of master integrals and present a brief review of algorithmic methods to solve reduction problems for Feynman integrals based on integration by parts relations. In particular, we discuss a recently suggested reduction algorithm which uses Groebner bases. New results obtained with its help for a family of three-loop Feynman integrals are outlined.