=ADD=
=reftype=
14
=number=
00-22
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/2000/00-22.ps.gz
=year=
2000
=month=
06
=author=
Perez-Diaz; S. + Schicho; J. + Sendra; J. R.
=title=
An Algorithmic Criterion for Deciding the Properness of Rational Parametrizations
=abstract=
In this paper we characterize the properness of rational parametrizations of
hypersurfaces by means of the existence of intersection points of some
additional algebraic hypersurfaces directly generated from the parametrization
over a field of rational functions. More precisely, if $V$ is a hypersurface over an algebraically closed field of characteristic zero and
${\cal P}(\overline{t})=\left(\frac{p_{1}(\overline{t})}{q_{1}(\overline{t})},
{\ldots},\frac{p_{n}(\overline{t})}{q_{n}(\overline{t})}\right)$ is a rational
parametrization of $V$, then the characterization is given in terms of the
intersection points of the hypersurfaces defined by
$x_{i}q_{i}(\overline{t})-p_{i}(\overline{t})$, ${i=1,{\ldots},n}$ over the
algebraic closure of ${\Bbb K}(V)$.
In addition, for the case of surfaces we show how these results can be stated
algorithmically. As a consequence we present an algorithmic criteria to decide
whether a given rational parametrization is proper that, in the affirmative
case, also computes the inverse of the parametrization. Moreover, for surfaces
the auxiliary hypersurfaces turn to be plane curves over ${\Bbb K}(V)$, and
hence the algorithm is essentially  based on resultants.
=sponsor=
DGES PB98-0713-C02-01, FWF project SFB013