Applying Gröbner Bases to Solve IBP Relations

Abstract

One more approach to solve reduction problems for Feynman integrals is based on the theory of Gröbner bases [56] that have arisen naturally when characterizing the structure of ideals of polynomial rings. The first attempt to apply this theory to Feynman integrals was made¹ in [202,204], where IBP relations were reduced to differential equations. To do this, one assumes that there is a non-zero mass for each line. The typical combination a_ii⁺, where i⁺is a shift operator, is naturally transformed into the operator of differentiation in the corresponding mass. Then one can apply some standard algorithms for constructing corresponding Gröbner bases for differential equations. Another attempt was made in [104] where Janet bases were used.