Parametrization Of Algebraic Curves Research Articles

Rationality and parametrizations of algebraic curves under specializations

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, image detection, algebraic differential equations, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a finer partition is possible such that the specialization of the parametrization parametrizes the specialized curve. Moreover, in this case, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified. We conclude the paper by illustrating these results by some concrete applications.

Parametrization of Algebraic Curves from a Number Theorist's Point of View

We present the technique of parametrization of plane algebraic curves from a number theorist's point of view and present Kapferer's simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions.

Parametrization of algebraic curves defined by sparse equations

We present a new method for the rational parametrization of plane algebraic curves. The algorithm exploits the shape of the Newton polygon of the defining implicit equation and is based on methods of toric geometry.

Parametrization of Algebraic Curves over Optimal Field Extensions

In this paper we investigate the problem of determining rational parametrizations of plane algebraic curves over an algebraic extension of least degree over the field of definition. This problem reduces to the problem of finding simple points with coordinates in the field of definition on algebraic curves of genus 0. Consequently we are also able to decide parametrizability over the reals. We generalize a classical theorem of Hilbert and Hurwitz about birational transformations. An efficient algorithm for computing such optimal parametrizations is presented.

The Parametrization of Algebraic Curves in Chiral Potts Models with Genus 1

The algebraic curves in chiral Potts models as and have been parametrized by using the method of parametrization algebraic curves with genus 1.