Computation of period matrices of real algebraic curves
This very well written paper explains a way to compute the period matrix of a real algebraic curve when it is given via uniformization, and not via equations. It describes an algorithm which has as input Möbius transformations freely generating a Fuchsian group \(G\) of the second kind, and as output the period matrix of the real curve \(X = \Omega (G)/G\), where \(\Omega(G)\) is the domain of discontinuity of \(G\). This accounts for real curves of genus \(g\) which separate their complexification, the case where the real curve does not separate may be reduced to the former using a double covering. The main idea is to express the holomorphic abelian differentials on \(X\) via Poincaré series, where the summation is over \(G\). The needed facts (convergence of Poincaré series, basis for holomorphic differentials for \(X\) given by Poincaré series easily obtained from the generators of \(G)\) are explained following closely a paper by \textit{C. J. Earle} and \textit{A. Marden} [Ill. J. Math. 13, 202-219 (1969; Zbl 0169.102)]. -- The author gives an estimation of the error made when the Poincaré series is truncated, and indicates that there are in this estimation constants which may be difficult to calculate from the presentation of \(G\). The paper closes with an example for genus 2, for which the computation time is given.
compute the period matrix of a real algebraic curve
scientific article; zbMATH DE number 502908
scientific article