Laureano Gonzalez-Vega , Departamento de Matematicas, Estadistica y Computacion, Universidad de Cantabria

Dealing with real algebraic curves symbolically and numerically for discovery: from experiments to theory
Geometric entities such as the set of the real zeros of a bivariate equation f(x,y)=0 can be treated algorithmically in a very efficient way by using a mixture of symbolic and numerical techniques. This implies that it is possible to know exactly which is the topology (connected components and their relative position, connectedness, singularities, etc.) of such a curve if the defining equation f(x,y) is also known in an exact manner (whatever this means) even for high degrees. In this talk we would like to describe three different "experiments" that highlight how new visualization tools in Computational Mathematics mixing symbolic and numerical techniques allow to perform experiments conveying either to mathematical discoveries and/or to new computational techniques. The first experiment to consider is not successful yet: it deals with the first part of Hilbert's sixteenth problem asking for the relative positions of the closed ovals of a non-singular algebraic curve f(x,y)=0 (regarded in the real projective plane). The problem is solved for degrees less or equal to 7 and, despite several attempts by H. Hong and F. Santos for degree 8, experimentation here has not produced yet any insight. Second experiment was successful: for a long period, in Algebraic Geodesy, it was not known where Vermeille’s method failed when inverting the transformation from geodetic coordinates to Cartesian coordinates. In 2009 L. Gonzalez-Vega and I. Polo-Blanco were able to compute exactly the region where Vermeille’s method cannot be applied by analysing carefully the topology of the arrangement of several real algebraic curves defined implicitly (coming from the solution of a quantifier elimination problem according to the terminology in Real Algebraic Geometry). Finally, third experiment has a different nature. In Computer Aided Design, the computation of the medial axis (or skeleton) of a region in the plane whose boundary is the union of finitely many curves has a wide range of applications. If the curves defining the region to compute the medial axis are restricted to be segments or conic arcs then I. Adamou, M. Fioravanti and L. Gonzalez-Vega have been able to determine all the possible “topologies” for the curves appearing in the medial axis (information which is specially useful when computing it). Again, the key tool here was the computation of the topology of the bisector curves (which in general are not rational) appearing in the medial axis to be computed.