Computing the Abel map

We present the next step in an ongoing research program to allow for the black-box computation of the so-called finite-genus solutions of integrable differential equations. This next step consists of the black-box computation of the Abel map from a Riemann surface to its Jacobian. Using a plane algebraic curve representation of the Riemann surface, we provide an algorithm for the numerical computation of this Abel map. Since our plane algebraic curves are of arbitrary degree and may have arbitrary singularities, the Abel map of any connected compact Riemann surface may be obtained in this way. This generality is necessary in order for these algorithms to be relevant for the computation of the finite-genus solutions of any integrable equation.

[1]  Mark van Hoeij,et al.  An Algorithm for Computing an Integral Basis in an Algebraic Function Field , 1994, J. Symb. Comput..

[2]  Bernard Deconinck,et al.  Computing Riemann matrices of algebraic curves , 2001 .

[3]  Bernard Deconinck,et al.  The KP equation with quasiperiodic initial data , 1998 .

[4]  Bernard Deconinck,et al.  Computing Riemann theta functions , 2002, Math. Comput..

[5]  Alexander I. Bobenko,et al.  All constant mean curvature tori inR3,S3,H3 in terms of theta-functions , 1991 .

[6]  P. Hacking,et al.  Riemann Surfaces , 2007 .

[7]  和達 三樹.  M. J. Ablowitz and H. Segur: Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, 1981, x+425ページ, 23.5×16.5cm, $54.40 (SIAM Studies in Applied Mathematics). , 1982 .

[8]  James H. Davenport,et al.  On the parallel Risch Algorithm (II) , 1985, TOMS.

[9]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[10]  Boris Dubrovin,et al.  Theta functions and non-linear equations , 1981 .

[11]  W. J. Harvey,et al.  TATA LECTURES ON THETA I (Progress in Mathematics, 28) , 1986 .

[12]  James H. Davenport,et al.  On the Integration of Algebraic Functions , 1979, Lecture Notes in Computer Science.

[13]  Michael F. Singer,et al.  Computing Galois Groups of Completely Reducible Differential Equations , 1999, J. Symb. Comput..

[14]  Phillip A. Griffiths,et al.  Introduction to Algebraic Curves , 1989 .

[15]  James H. Davenport.  The Parallel Risch Algorithm (I) , 1982, EUROCAM.

[16]  D. Mumford.  Tata Lectures on Theta I , 1982 .

[17]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[18]  M. H. Protter,et al.  THE SOLUTION OF THE PROBLEM OF INTEGRATION IN FINITE TERMS , 1970 .

[19]  R. Risch.  The problem of integration in finite terms , 1969 .

[20]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[21]  Allan P. Fordy,et al.  Soliton Theory: A Survey of Results , 1990 .

[22]  Ljudmila A. Bordag,et al.  Periodic multiphase solutions of the Kadomtsev-Petviashvili equation , 1989 .

[23]  H. Piaggio.  Algebraic Functions , 1952, Nature.

[24]  E. V. Flynn.  Large rational torsion on Abelian varieties , 1990 .