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\title{A. Weil witness variety construction for parametric curves
\thanks{Joint work with C. Andradas and J.R. Sendra. An
extended version will appear in the Procedings of ISSAC
99. Supported by DGESIC 1FD-97-0409 ``T\'ecnicas
algebro-num\'ericas en Dise\~no Geom\'etrico asistido por
Computador" } }
\author{ Tom\'as Recio\\
Universidad de Cantabria.\\
{\tt recio@matesco.unican.es}}
\date{}
\maketitle
\noindent {\bf {\large 1-}} Suppose we
are given a parametric curve $V$ through a rational parametric mapping
with coefficients in some algebraic separable extension $k'$ of a
computable field $k$. The problems we will like to deal with in the
talk are
\begin{enumerate}
\item[1)] to decide whether $V$ is defined as the zero
set of some polynomial equations with coefficients in $k$,
\item[2)] to decide whether $V$ has an infinite number of
$k$-rational points (i.e. whether $V$ has a $k$-rational
parametrization).
\end{enumerate}
\noindent Both problems are classical (see \cite{ARS}
for a more detailed presentation and the references thereof), but they
are usually treated considering as input data a collection of implicit
equations for $V$. For implicitly presented curves, computing any
Gr\"obner basis of the curve's ideal and collecting the coefficients
of the polynomials therein, is enough to solve question 1), since
these coefficients will generate (over $k$) the field of definition
for $V$. On the other hand, finding an optimal parametrization for $V$
can be achieved via adjoint curves (\cite{SW}) or canonical divisors
(\cite{H}). In fact, if the curve is definable over $k$, then its
function field is $k$-birrational to the function field of a
$k$-conic, and thus it can be parametrized over an extension of $k$ of
degree at most two.
\noindent {\bf {\large 2-}} But if the curve is parametrically
given, it seems aesthetically unpleasant and computationally expensive
to implicitize in order to extract information that it is already
contained in the function field of $V$ over $k$, which is completely
known after the parametric equations are given. In fact, this
philosophy is followed also in our previous work
\cite {ARS} but there the goal was to
obtain, via the canonical divisor, the optimal parametrization of $V$
(assuming it is definable over $k$). In this talk the addressed
problem is just a decision one, and accordingly, the solution is
simpler but conceptually quite surprising. Given a variety $V$,
implicitly defined over an algebraic separable field extension
$k(\alpha)$, A. Weil
\cite {Weil} developed a restriction technique (called by him a {\it
descente} method), that associates
to $V$ a suitable $k$-variety $W$, such that many properties of $V$ can be
analyzed by merely looking at
$W$, that is, by descending to the base field $k$. Because of this, we
think of this variety $W$ as a ``witness" for $V$.
\noindent {\bf {\large 3-}} In the talk we will
present a parametric counterpart, for curves, of Weil's
construction. As an application, we can state some simple
algorithmic criteria over the variety
$W$ that translate, for instance, the $k$-definability of a parametric
curve $V$, or the existence of an infinite number of
$k$-rational points in
$V$. In fact, it turns out that, by
adapting Weil's construction to the parametric case, we have been
able to identify an algebraic set $U$ defined over $k$ and of
dimension less or equal to one (that could be called the
``proper" restriction of $V$
to the base field $k$) that allows
solving the above mentioned problems. For instance, we will prove
that
$V$ is definable over $k$ iff $U$ is a curve and that
$V$ is parametrizable over $k$ iff $U$ is a hypercircle.
Moreover, parametrizing this hypercircle yields a
$k$-parametrization of
$V$. Finally, roughly speaking, this hypercircle turns to be a line
iff
$V$ admits a polynomial parametrization over $k$.
{\small
\begin{thebibliography}{}
\bibitem{ARS} Andradas, C.; Recio, T.; Sendra, R. (1997)
{\it
Relatively Optimal Rational Space
Curve Reparametrization Algorithm through Canonical
Divisors}.
Proc. ISSAC 97. pp. 349-355, W. W. K\"uchlin, Ed. ACM press.
\bibitem{H} van Hoeij, M. (1997) {\it Rational parametrization
of curves using canonical divisors}, JSC, vol. 23.
\bibitem{SW} Sendra, R.; Winkler, F. (1997) {\it Parametrization
of algebraic curves over optimal field extensions}, JSC, vol. 23.
\bibitem{Weil} Weil, A. (1959) {\it Adeles et groupes algebriques,}
Seminaire Bourbaki, no. 186.
\end{thebibliography}
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