Date: 21 Dec 2010

Algebraic Curves and Riemann Surfaces in Matlab

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Abstract

In the previous chapter, a detailed description of the algorithms for the ‘algcurves’ package in Maple was presented. As discussed there, the package is able to handle general algebraic curves with coefficients given as exact arithmetic expressions, a restriction due to the use of exact integer arithmetic. Coefficients in terms of floating point numbers, i.e., the representation of decimal numbers of finite length on a computer, can in principle be handled, but the floating point numbers have to be converted to rational numbers. This can lead to technical difficulties in practice. One also faces limitations if one wants to study families of Riemann surfaces, where the coefficients in the algebraic equation defining the curve are floating point numbers depending on a set of parameters, i.e., if one wants to explore modular properties of Riemann surfaces as in the examples discussed below. An additional problem in this context can be computing time since the computation of the Riemann matrix uses the somewhat slow Maple integration routing. Thus, a more efficient computation of the Riemann matrix is interesting if one wants to study families of Riemann surfaces or higher genus examples which are computationally expensive.