Gianni, Patrizia; Seppälä, Mika; Silhol, Robert; Trager, Barry Riemann surfaces, plane algebraic curves and their period matrices. (English) Zbl 0964.14047 J. Symb. Comput. 26, No. 6, 789-803 (1998). Summary: The aim of this paper is to present a theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program CARS [K.-D. Semmler, T. Ventjoki, J. Levinen, L. Renggli and M. Seppälä, “CARS. Program for defining Riemann surfaces for computations” (1993-1996)] that can be used to define Riemann surfaces 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 complex analysis and algebraic geometry. The contribution of the present paper is the design of an algorithm which is based on the classical results and computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve. This algorithm thus solves an important problem in the general case. This problem was first solved, in the case of symmetric Riemann surfaces, by M. Seppälä [Discrete Comput. Geom. 11, No. 1, 65-81 (1994; Zbl 0805.14029)]. Cited in 1 ReviewCited in 8 Documents MSC: 14Q15 Computational aspects of higher-dimensional varieties 30F10 Compact Riemann surfaces and uniformization 14H55 Riemann surfaces; Weierstrass points; gap sequences 32G20 Period matrices, variation of Hodge structure; degenerations 14-04 Software, source code, etc. for problems pertaining to algebraic geometry MathOverflow Questions: How can I calculate the period matrix of this Riemann surface? Keywords:computing a representation of a compact Riemann surface; period matrix; CARS Software:CARS PDF BibTeX XML Cite \textit{P. Gianni} et al., J. Symb. Comput. 26, No. 6, 789--803 (1998; Zbl 0964.14047) Full Text: DOI