Computing Differential Equations for Integrals Associated to Smooth Fano Polytopes

TL;DR

This paper presents an approximate algorithm for computing differential equations related to integrals of smooth Fano polytopes, with applications to K3 surfaces and mirror symmetry, providing a finite-step solution under certain conditions.

Contribution

The paper introduces a finite-step approximate algorithm for holonomic systems of differential equations associated with integrals of smooth Fano polytopes, leveraging Stienstra's rank formula for stopping criteria.

Findings

Algorithm correctly computes differential equations in finite steps.

Application to smooth Fano polytopes relevant to K3 surfaces.

Utilizes Stienstra's rank formula for algorithm termination.

Abstract

We give an approximate algorithm of computing holonomic systems of linear differential equations for definite integrals with parameters. We show that this algorithm gives a correct answer in finite steps, but we have no general stopping condition. We apply the approximate method to find differential equations for integrals associated to smooth Fano polytopes. They are interested 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 approximate algorithm.