Fields of Parametrization and Optimal Affine Reparametrization of Rational Curves

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dc.creatorTabera, Luis Felipe
dc.date2008-10-30
dc.date.accessioned2026-07-07T10:14:31Z
dc.date.available2026-07-07T10:14:31Z
dc.descriptionIn this paper we present three related results on the subject of fields of parametrization. Let C be a rational curve over a field of characteristic zero. Let K be a field finitely generated over Q, such that it is a field of definition of C but not a field of parametrization. It is known that there are quadratic extensions of K that parametrize C. First, we prove that there are infinitely many quadratic extensions of K that are fields of parametrization of C. As a consequence, we prove that the witness variety, that appear in the context of the parametric Weil's descente method, is always a special curve related to algebraic extensions, called hypercircle. It is possible that the witness variety is not a hypercircle for the given extension, but for an alternative one. We use these two facts to present an algorithm to solve the following optimal reparametrization problem. Given a birational parametrization f(t) of a curve C, compute the affine reparametrization at+b such f(at+b) has coefficients over a field as small as possible. The main advantage of this algorithm is that it does not need to compute any rational point on the curve.
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/0810.5595
dc.identifierhttp://arxiv.org/abs/0810.5595
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/172905
dc.subjectAlgebraic Geometry
dc.subjectCommutative Algebra
dc.subject14H99 14M20
dc.titleFields of Parametrization and Optimal Affine Reparametrization of Rational Curves
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