Computation of Gröbner Bases for Two-Loop Propagator Type Integrals

The most efficient algorithms to evaluate Feynman diagrams exploit recursive methods based on integration by parts technique [2] or technique of generalized recurrence relations [3], [4]. It is easy to derive hundreds of recurrence relations but it is by far not so easy to formulate optimal algorithm how to use them for the reduction of a given type of integrals to a minimal set of basis integrals. In [1] it was proposed to use the Gröbner basis method [5] as a mathematical background to solve this problem. This proposal can be realized in different ways. For example, the system of recurrence relations can be rewritten as a system of partial differential equations (PDE). The set of relations needed for reduction of Feynman integrals with different powers of propagators to the set of basis integrals will be Gröbner basis of this overdetermined system of differential equations. Information about the minimal basis of integrals also will be contained in the Gröbner basis. Another possibility will be to rewrite recurrence relations in terms of operators shifting powers of propagators. Each equation should be considered as an operator. Operators shifting powers of propagators satisfy an Ore algebra’ condition and therefore, to find minimal set of recurrence relations from our system of equations ammounts to computation of it’s Gröbner basis in Ore algebra. Our preliminary investigation reveals that both approaches have some merits and shortcomings.