Solving bivariate algebraic systems and computing the toplogy of curves

par Yacine Bouzidi (INRIA Lille)

jeudi 11 févr. 2016, 10:30 → 11:30 Europe/Paris

Salle XR203 (Bâtiment XLIM)

A fundamental problem in computational geometry is the computation of the topology of an algebraic plane curve given by its implicit equation, that is, the computation of a graph lines that approximates the curve while preserving its topology. A critical step in many algorithms computing the topology of a plane curve is the computation of the set of singular and extreme points (wrt x) of this curve, which is equivalent to the computation of the solutions of bivariate systems defined by the curve and some of its partial derivatives. In this presentation, we study form theoretical and practical perspectives the problem of solving systems of bivariate polynomials with integer coefficients. More precisely, we investigate the computation of a Rational Univariate Representation (RUR) of the solutions of a bivariate system, that is, a one-to-one mapping that sends the roots of a univariate polynomial to the solutions of the bivariate system. We also show some applications of our results in the context of dynamical system theory.