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Bounding the Betti numbers of real tropical varieties. Renaudineau, Arthur


Real tropical varieties are polyhedral objects which are in some cases isotopic to real algebraic varieties. We will introduce those objects and show an upper bound on their Betti numbers. These bounds are given in terms of the dimension of tropical homology groups modulo 2 of the underlying tropical variety, and are derived from a certain spectral sequence. In the case of hypersurfaces, we prove that tropical homology groups are torsion free, implying a bound conjectured by Itenberg on the Betti numbers of the real part of a real hypersurface near the tropical limit in terms of Hodge numbers of the complexification. This is a joint work with Kristin Shaw, and with Charles Arnal and Kristin Shaw.

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