Séminaire de Géométrie Tropicale
Mercredi 30 septembre 2015 salle 1525-502
Polynomial systems with many positive solutions from bipartite triangulations
Résumé :
We use a version of Viro's method to construct polynomial systems with many positive solutions.
We show that if a polytope admits an unimodular regular triangulation whose dual graph is bipartite, then there exists
an unmixed polynomial system with this polytope as Newton polytope and which is maximally positive in that all its toric complex solutions
are in fact real positive solutions. We present classical families of polytopes which admit such triangulations. These examples give evidence in favor of a conjecture due to Bihan which characterizes affine relations inside the support of a maximally polynomial system. We also use our construction to get polynomial systems with many positive solutions considering a simplicial complex contained in a regular triangulation of the cyclic polytope. This is joint work with Pierre-Jean Spaenlehauer (INRIA Nancy).
Fake real planes: exotic affine algebraic models of R²
Résumé :
We study topologically minimal complexifications of the euclidean affine plane R² up to isomorphisms and up to birational diffeomorphisms.
A fake real plane is a smooth geometrically integral surface S defined over R such that:
The real locus S(R) is diffeomorphic to R²;
The complex surface S_C(C) has the rational homology type of A²_C.;
S is not isomorphic to A²_R as surfaces defined over R.
The analogous study in the compact case, that is the classification of complexifications of the real projective plane P²(R) with the rational homology of the complex projective plane is well known: P²_C is the only one.
We prove that fake real planes exist by giving many examples and we tackle the question: does there exist fake planes S such that S(R) is not birationally diffeomorphic to A²_R(R)?
(Joint work with Adrien Dubouloz.)
Anneau de Chow combinatoire d'un produit de graphes
Résumé :
Je décrirai la structure combinatoire d'un anneau de Chow associé aux produits de graphes, qui contrôle le comportement analytique des produits d'intersection locaux dans un produit de courbes sur un corps non-archimédien d'après les travaux de Shou-Wu Zhang et Johannes Kolb.