Rigorous numerical integration of algebraic functions

TL;DR

This paper introduces a rigorous numerical integration algorithm for algebraic functions using Fujiwara's inequality, with practical improvements via path splitting, demonstrated through computing period matrices in SageMath.

Contribution

It presents a new algorithm combining Fujiwara's inequality with path splitting for accurate algebraic function integration, implemented in SageMath.

Findings

Effective bounds for algebraic functions over ellipses

Path splitting significantly improves convergence

Successfully computed period matrices of algebraic Riemann surfaces

Abstract

We present an algorithm which uses Fujiwara's inequality to bound algebraic functions over ellipses of a certain type, allowing us to concretely implement a rigorous Gauss-Legendre integration method for algebraic functions over a line segment. We consider path splitting strategies to improve convergence of the method and show that these yield significant practical and asymptotic benefits. We implemented these methods to compute period matrices of algebraic Riemann surfaces and these are available in SageMath.