Riemann surfaces, plane algebraic curves and their period matrices

The aim of this paper is to present theoretical basis for computing a repre sentation of a compact Riemann surface as an algebraic plane curve and to c ompute a numerical approximation for its period matrix. We will describe a program CARS (Semmler ct al., 1996) that can be used to define Riemann surf aces for computations. CARS allows one also to perform the Fenchel-Nielsen twist and other deformations on Riemann surfaces. Almost all theoretical results presented here are well known in classical c omplex analysis and algebraic geometry. The contribution of the present pap er is the design of an algorithm which is based on the classical results an d computes first an approximation of a polynomial representing a given comp act Riemann surface as a plane algebraic curve and further computes an appr oximation for a period matrix of this curve. This algorithm thus solves an important problem in the general case. This problem was first solved, in th e case of symmetric Riemann surfaces, in Seppala (1994). (C) 1998 Academic Press.