Advances in Algebraic Geometric Computation

Algebraic curves and surfaces play an important and ever increasing role in computer aided geometric design, computer vision, and computer aided manufacturing. Consequently, theoretical results need to be adapted to practical needs. We need efficient algorithms for generating, representing, manipulating, analyzing, rendering algebraic curves and surfaces. In the last years there has been dramatic progress in all areas of algebraic computation. In particular, the application of computer algebra to the design and analysis of algebraic curves and surfaces has been extremely successful. In this lecture we report on some of these developments. One interesting subproblem in algebraic geometric computation is the rational parame-trization of curves and surfaces. The tacnode curve defined by f(x, y) = 2x − 3xy + y − 2y + y in the real plane has the rational parametrization x(t) = t − 6t + 9t− 2 2t4 − 16t3 + 40t2 − 32t + 9 , y(t) = t − 4t + 4 2t4 − 16t3 + 40t2 − 32t + 9 over Q. The criterion for parametrizability is the genus. Only curves of genus 0 have a rational parametrization, and only surfaces of arithmetic genus 0 and second plurigenus 0 have a rational parametrization. Conversely, given a parametric representation of a curve or surface, we might ask for the implicit algebraic equation defining it. Computing parametrizations essentially requires the full analysis of singularities (either by successive blow-ups, or by Puiseux expansion) and the determination of regular points on the curve or surface. We can control the quality of the resulting parametrization by controlling the field over which we choose this regular point. Thus, finding a regular curve point over a minimal field extension on a ∗The author wants to acknowledge support from the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under project SFB F013/1304. 108 Proceedings of the International Conference on Algebra and Its Applications 2002 curve of genus 0 is one of the central problems in rational parametrization of curves, compare [SeWi97]. Similarly, finding rational curves on surfaces leads to parametrizations, compare [LSWH00]. The quality of parametrizations can be measured by the necessary field extension and also by the number of times the variety is traced by the parametrization. We will analyze the relation of the tracing index of a curve to the degrees of the implicit equation and the degree of the parametrization, compare [SeWi01]. 1 Parametrization of Algebraic Curves Algebraic curves and surfaces have been studied intensively in algebraic geometry for decades and even centuries. Thus, there exists a huge amount of theoretical knowledge about these geometric objects. Recently, algebraic curves and surfaces play an important and ever increasing role in computer aided geometric design, computer vision, and computer aided manufacturing. Consequently, theoretical results need to be adapted to practical needs. We need efficient algorithms for generating, representing, manipulating, analyzing, rendering algebraic curves and surfaces. Such efficient symbolic algorithms can be constructed based on method of computer algebra as described, for instance, in [Wink96]. One interesting subproblem is the rational parametrization of curves and surfaces. Definition 1.1. Let K be an algebraically closed field of characteristic 0. Consider an affine plane algebraic curve C in A(K) defined by the bivariate polynomial f(x, y) ∈ K[x, y], i.e. C = {(a, b) | (a, b) ∈ A(K) and f(a, b) = 0}. Of course, we could also view this curve in the projective plane P(K), defined by F (x, y, z), the homogenization of f(x, y). We denote the field of rational function over C by K(C). A pair of rational functions P = (x(t), y(t)) ∈ K(t) is a rational parametrization of the curve C, if and only if f(x(t), y(t)) = 0 and for almost every point (x0, y0) ∈ C (i.e. up to finitely many exceptions) there is a parameter value t0 ∈ K such that (x0, y0) = (x(t0), y(t0)). Only irreducible curves, i.e. curves whose defining polynomials are absolutely irreducible, can have a rational parametrization. Almost any rational transformation of a rational parametrization is again a rational parametrization, so such parametrizations are not unique. Implicit representations (by defining polynomial) and parametric representations (by rational parametrization) both have their particular advantages and disad-