Efficient computation of the branching structure of an algebraic curve

TL;DR

This paper introduces an efficient algorithm for determining the branching structure of algebraic curves by constructing minimal spanning trees and analytically continuing roots, facilitating period computations of Riemann surfaces.

Contribution

The paper presents a novel method combining minimal spanning trees and root continuation to efficiently compute the branching structure of algebraic curves.

Findings

Algorithm reduces computational complexity for branching analysis.

Constructs minimal spanning trees for discriminant points.

Enables accurate period calculations of Riemann surfaces.

Abstract

An efficient algorithm for computing the branching structure of a compact Riemann surface defined via an algebraic curve is presented. Generators of the fundamental group of the base of the ramified covering punctured at the discriminant points of the curve are constructed via a minimal spanning tree of the discriminant points. This leads to paths of minimal length between the points, which is important for a later stage where these paths are used as integration contours to compute periods of the surface. The branching structure of the surface is obtained by analytically continuing the roots of the equation defining the algebraic curve along the constructed generators of the fundamental group.