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Séminaire Géométrie et théorie des modèles

Horaires : Le vendredi 15 décembre 2017, 16h

Lieu : ENS. salle W

The geometry of combinatorially extreme algebraic configurations

Martin Bays (Muenster)

Given a system of polynomial equations in m complex variables with solution set of dimension d, if we take finite subsets X_i of C each of size at most N, then the number of solutions to the system whose ith co-ordinate is in X_i is easily seen to be bounded as O(N^d). We ask: when can we improve on the exponent d in this bound? Hrushovski developed a formalism in which such questions become amenable to the tools of model theory, and in particular observed that incidence bounds of Szemeredi-Trotter type imply modularity of associated geometries. Exploiting this, we answer a (more general form of) our question above. This is part of a joint project with Emmanuel Breuillard.

 

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