Computer-assisted proofs for finding the monodromy of Picard-Fuchs differential equations for a family of K3 toric hypersurfaces

TL;DR

This paper introduces a rigorous numerical method to compute the monodromy of Picard-Fuchs differential equations, demonstrated on K3 toric hypersurfaces, combining interval arithmetic and analytic continuation for precise results.

Contribution

It presents a novel computer-assisted approach for rigorously determining monodromy matrices of differential equations, specifically applied to K3 toric hypersurfaces.

Findings

Successfully computed monodromy matrices for a K3 hypersurface example.

Established a rigorous numerical framework for monodromy problems.

Validated the method's accuracy with interval arithmetic.

Abstract

In this paper, we present a numerical method for rigorously finding the monodromy of linear differential equations. Beginning at a base point where certain particular solutions are explicitly given by series expansions, we first compute the value of fundamental system of solutions using interval arithmetic to rigorously control truncation and rounding errors. The solutions are then analytically continued along a prescribed contour encircling the singular points of the differential equation via a rigorous integrator. From these computations, the monodromy matrices are derived, generating the monodromy group of the differential equation. This method establishes a mathematically rigorous framework for addressing the monodromy problem in differential equations. For a notable example, we apply our computer-assisted proof method to resolve the monodromy problem for a Picard--Fuchs differential equation associated with a family of K3 toric hypersurfaces.