Computational approach to compact Riemann surfaces

TL;DR

This paper introduces a numerical method for analyzing compact Riemann surfaces derived from algebraic curves, utilizing Newton iteration, spectral methods, and contour integration to compute fundamental properties with high accuracy.

Contribution

It presents a novel numerical framework combining Newton iteration, spectral collocation, and contour integration for the analysis of compact Riemann surfaces from algebraic curves.

Findings

Accurate computation of critical points and monodromies.

Spectral accuracy in period calculations using Chebyshev collocation.

Application to solutions of the Kadomtsev-Petviashvili equation.

Abstract

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. As an application of the code, solutions to the Kadomtsev-Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.