Issues in Symbolic Parametrization of Algebraic Curves ^{∗}

Franz Winkler Institut für Mathematik and RISC-LINZ Johannes Kepler Universität A-4040 Linz, Austria

                       e-mail: winkler@risc.uni-linz.ac.at

Abstract

The problem of rational parametrization of plane algebraic curves is best explained by an example. Suppose we consider the curve C_{1} in the complex plane, whose points are the zeros of the polynomial

f_{1}(x, y) = (x^{2} + 4y + y^{2})^{2} − 16(x^{2} + y^{2}).

C_{1} is a curve of genus 0, so it can be rationally parametrized, e.g. as

x(t) = \frac{9216t^{4}−62464t^{3}+141696t^{2}−108864t+2916}{430336t^{4}−94464t^{3}+58976t^{2}−5904t+1681} ,
y(t) = \frac{40960t^{4}−184320t^{3}+204800t^{2}+11520t−12960}{430336t^{4}−94464t^{3}+58976t^{2}−5904t+1681} .

In general we say that a curve C has a rational parametrization, iff there are (nontrivial) rational functions x(t), y(t) such that the defining polynomial f of C vanishes on them,

f(x(t), y(t)) = 0.

The parametrizable curves are exactly the irreducible curves of genus 0, i.e. they achieve the upper bound on the number of singularities. If such a parametrization exists, then it is by no means unique, even if we require that the degrees of the rational functions involved are minimal. In fact, one of the main problems is to determine a minimal extension of the field of definition, in which a parametrization can be expressed.

For this purpose we assume that K is a computable field of characteristic 0, we call it the field of definition. \overline{K} denotes the algebraic closure of K. The defining polynomial of the curve C that we want to parametrize has coefficients in K. The curve C itself, however, is considered to be a curve over \overline{K}. In the parametrization process we might have to extend K algebraically. We will ultimately construct a parametrization (if one exists) over some field K(γ) ⊆ \overline{K}, where γ is algebraic over K. Of course, we want to keep the degree of γ as low as possible.

Singular points on algebraic curves occur only in full conjugacy classes. Consequently they do not introduce algebraic numbers in the parametrization process. In general, however, we also need a certain number of simple points on the curve. This might force us to extend the field of definition of the curve, in order to arrive a field over which we can give a parametrization.

From the work of Nöther and Hilbert and Hurwitz we know that it is possible to parametrize any curve C of genus 0 over the field of definition K, if deg(C) is odd, and over some quadratic extension of K, if deg(C) is even. In a previous paper J.R. Sendra and the author have presented an algorithm which actually achieves this optimal field of parametrization. Moreover, if the field of definition is the field of rational numbers, we can also decide if the curve can be parametrized over the reals, and compute such a parametrization if it exists.

Our parametrization algorithm is of polynomial bit complexity if we assume that the curve C defined over the rational numbers has only ordinary singularities. If we allow curves with also non–ordinary singularities, then the bit complexity becomes exponential in the worst case.

Keywords: parametrization of algebraic curves, rational points on algebraic curves, complexity.

∗Supported by Österr. Fonds zur Förderung der wissenschaftlichen Forschung, Proj. POSSO, Nr. P9181-TEC. (ESPRIT BRA No. 6846).