SPEAKER: Xavier Allamigeon (Paris)

TITLE: Tropicalization of spectrahedra

ABSTRACT: We introduce tropical spectrahedra, defined as the images by the nonarchimedean valuation of spectrahedra over the field of real Puiseux series. We provide an explicit characterization of generic tropical spectrahedra, involving principal tropical minors of size at most 2. To do so, we study the images by the nonarchimedean valuation of semialgebraic sets. We prove in particular that, under a regularity assumption, the image by the valuation of a basic semialgebraic set is obtained by tropicalizing the inequalities which define it. We finally show that the projections of tropical spectrahedra are precisely the sets of winning certificates of stochastic mean-payoff games, and discuss the applications of these results to semidefinite programming over nonarchimedean fields. This is joint work with Stéphane Gaubert and Mateusz Skomra.