Computational approach to compact Riemann surfaces. (English) Zbl 1360.14142
This paper gives a computational approach to compact Riemann surfaces, starting from plane algebraic curves. The main methods and results are listed as follows.
1) Using a two-dimensional Newton iteration, the critical points of the algebraic curves (i.e. branch points and singular points) can be computed. The starting points for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros.
2) A set of generators of the fundamental group of the complex plane minus the critical points of the algebraic curve, is constructed from circles around these points and connecting lines obtained from a minimal spanning tree.
3) Monodromies, are obtained by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. Collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials. This gives the periods of the Riemann surface with spectral accuracy.
4) Holomorphic differentials are identified using Puiseux expansions at the singular points of the algebraic curve. These Puiseux expansions are computed by contour integration on the circles around the singularities.
5) The Abel map is also determined using the Clenshaw-Curtis algorithm and contour integrals.
Finally, as an application of the code, one obtains solutions to the Kadomtsev-Petviashvili equation on non-hyperelliptic Riemann surfaces.
1) Using a two-dimensional Newton iteration, the critical points of the algebraic curves (i.e. branch points and singular points) can be computed. The starting points for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros.
2) A set of generators of the fundamental group of the complex plane minus the critical points of the algebraic curve, is constructed from circles around these points and connecting lines obtained from a minimal spanning tree.
3) Monodromies, are obtained by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. Collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials. This gives the periods of the Riemann surface with spectral accuracy.
4) Holomorphic differentials are identified using Puiseux expansions at the singular points of the algebraic curve. These Puiseux expansions are computed by contour integration on the circles around the singularities.
5) The Abel map is also determined using the Clenshaw-Curtis algorithm and contour integrals.
Finally, as an application of the code, one obtains solutions to the Kadomtsev-Petviashvili equation on non-hyperelliptic Riemann surfaces.
Reviewer: Noémie Combe (Marseille)
MSC:
| 14Q05 | Computational aspects of algebraic curves |
| 14H45 | Special algebraic curves and curves of low genus |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 14H10 | Families, moduli of curves (algebraic) |
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