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Tropical spectrahedra

Xavier Allamigeon 1, 2 Stéphane Gaubert 1, 2 Mateusz Skomra 1, 2 
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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 show that the nonarchimedean valuation maps semialgebraic sets to semilinear sets that are closed. We also prove that, under a regularity assumption, the image by the valuation of a basic semialgebraic set is obtained by tropicalizing the inequalities which define it.
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Submitted on : Monday, December 26, 2016 - 3:21:39 PM
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Xavier Allamigeon, Stéphane Gaubert, Mateusz Skomra. Tropical spectrahedra. Discrete and Computational Geometry, Springer Verlag, 2020, 63, pp.507-548. ⟨10.1007/s00454-020-00176-1⟩. ⟨hal-01422639⟩



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