Computational approach to compact Riemann surfaces
Computational approach to compact Riemann surfaces
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紧致黎曼曲面的计算方法
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
C. Klein
中科院分区：
文献类型：
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作者：
J. Frauendiener;C. Klein
从平面代数曲线出发，给出了紧致黎曼曲面的一种纯数值方法。通过二维牛顿迭代计算的代数曲线的临界点。该迭代的起始值是从关于代数曲线的坐标和它们的零点的适当配对的结式获得的。复平面上这些临界点的补的基本群的一组生成元是由这些点周围的圆和从最小生成树获得的连接线构造的。单值线的计算是通过求解沿着这些轮廓上的配点上的代数曲线的定义方程和解析地连续根来进行的。配置点的选择，以对应于Chebychev配置点，为随后的Clenshaw-Curtis积分的全纯微分给出的周期的黎曼曲面的频谱精度。在代数曲线的奇点处，通过在奇点周围的圆上进行围线积分计算出的Puixix展开式来识别全纯微分。Abel映射也用Clenshaw-Curtis算法和轮廓积分计算。作为该程序的一个应用，Kadomtsev-Petviashvili方程的解在非超椭圆Riemann曲面上进行了计算。
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.