Computing differential equations for integrals associated to smooth Fano polytope

Abstract

We give an approximation algorithm of computing holonomic systems of linear differential equations for definite integrals of rational functions with parameters. We show that this algorithm gives a correct answer in finite steps, but we have no general stopping condition. We apply the approximation method to find differential equations for integrals associated to smooth Fano polytopes. These are interesting in the study of K3 surfaces and the toric mirror symmetry. In this class of integrals, we can apply Stienstra’s rank formula to our algorithm, which gives a stopping condition of the approximation algorithm.