Computing Riemann matrices of algebraic curves

A black-box program for the explicit calculation of Riemann matrices of arbitrary compact connected Riemann surfaces is presented. All such Riemann surfaces are represented as plane algebraic curves. These algebraic curves are allowed to have arbitrary singularities. The method of calculation of the Riemann matrix is essentially its deenition: we numerically integrate the holomorphic diierentials of the Riemann surface over the cycles of a canonical basis of the homology of the Riemann surface. Both the holomorphic diierentials and the canonical basis of the homology of the Riemann surface are obtained exactly through symbolic calculations. This program is included in MapleV.6, as part of the algcurves package. 1 Motivation Integrable partial diierential equations such as the Korteweg-deVries (KdV) equation and the Nonlinear Schrr odinger (NLS) equation have been widely used in the last thirty years for the description of various physical phenomena. Especially the soliton solutions of these equations have claimed a well-deserved niche in both theory and experiment. It is however fair to say that their popularity has not been shared with their periodic and quasiperiodic counterparts. There are many issues one could point at to explain this discrepancy, but the main one must be that the theory of the periodic and quasiperiodic solutions of these equations invariably is connected with algebraic geometry and the theory of Riemann surfaces 2, 8]. If one manages to obtain explicit formulas for the solutions, they usually involve Riemann theta functions, parametrized by some Riemann surface. This lack of explicitness should be compared with the usually much simpler explicit formulas one obtains for one-and two-soliton solutions, in terms of exponential and rational functions. A Riemann theta function with g phases is given by