##
Numerical uniformization of hyperelliptic curves

Appeared in Proceedings ISSAC95

### Abstract

We develop algorithms
which allow us to uniformize numerically
any given
real hyperelliptic plane M-curve.
Starting from an equation for such a curve, we
get (floating point approximations) for the generators of
a discontinuous group of Möbius transformations uniformizing
the given M-curve.
Furthermore, we show how to compute an equation for
a Riemann surface given by a such a group.
For real hyperelliptic plane curves having
maximal number of real components this construction gives a
complete answer to the problem of numerical uniformization.

Supported by the EC HCM
network``Computational Conformal Geometry'', Human Capital
and Mobility plan contract # CHRX-CT93-0408,
Dec. 1, 1993 - Nov. 30, 1996.

September 19, 1995
M. Seppälä