Séminaire de Géométrie Tropicale
13 mars 2013, 16h15 salle 1525-502
Johannes Rau
(Universität des Saarlandes)
Résumé :
Topological classification of real rational curves in the plane
The study of the topological shape of real algebraic curves in the plane
has a long history. Research was focused mostly on smooth curves, for which
Viro's patchworking method provides a tool to construct curves with
specific topological properties. Tropical geometry generalizes patchworking
in so far as it allows to patchwork also (certain) singular curves. In a
joint work with Ilia Itenberg and Grigory Mikhalkin, we used tropical
construction methods to classify rational curves up to degree 5. In
particular, for degree 5 and if all the nodes of the curve are
intersections of two real branches, there are 40 possible topological
types. They can all be realized in an easy manner using only three tropical
curves.