DK9: Symbolic-Numeric Techniques for Genus Computation and Parametrization

A plane algebraic curve, given by a polynomial equation in 2 variables, is parametrizable by rational functions if and only if its genus is zero. For polynomials with real or complex coefficients, both the genus computation problem and the parametrization problem are severely ill-posed: a slight change in the coefficients may change the genus (which is always an integer) and may turn a parametrizable curve into a non-parametrizable one.

The goal of the proposed PhD project is to give symbolic-numerical algorithms that compute the genus, and if applicable to compute an approximate rational parametrization, in the sense of approximate algebraic computation: The computed genus is the lowest genus of a curve with nearby coefficients, and if the computed genus is zero, then the computed parametrization is the exact parametrization of a nearby rational curve.

In exact symbolic algorithms, the genus is computed via an analysis of the singularities, either by Puiseux expansion or by a resolution. In the complex case, the singularities can be characterized by their link, which is defined as the intersection of the curve with a 3-sphere with sufficiently small radius. When this radius is chosen carefully, one can compute the topological type of the link in a numerically stable way. The main idea is to combine these informations on the singularities and the knowledge on the shape of the defining equation (degree, Newton polygon) in order to give an estimation of the genus in the above sense, which also decides the existence of an approximate parametrization.

Supervisor: Prof. Josef Schicho