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Tropical Spectrahedra

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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 polyhedral characterization of generic tropical spectrahedra, involving principal tropical minors of size at most 2. One of the key ingredients is Denef–Pas quantifier elimination result over valued fields. We obtain from this 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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Notes

  1. Note that the dependence on D lies here, as the choice of D restricts the amount of possible functions \(e\mapsto i_{e}\), \(e\mapsto (i_{e}, j_{e})\).

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Acknowledgements

We thank the referees for their comments which led to improvements of this article.

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Authors and Affiliations

  1. INRIA and CMAP, École Polytechnique, IP Paris, CNRS, 91128, Palaiseau Cedex, France

    Xavier Allamigeon & Stéphane Gaubert

  2. Université de Lyon, EnsL, UCBL, CNRS, LIP, 69342, Lyon Cedex 07, France

    Mateusz Skomra

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  1. Xavier Allamigeon
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Correspondence to Mateusz Skomra.

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Xavier Allamigeon, Stéphane Gaubert, and Mateusz Skomra were partially supported by the ANR projects CAFEIN (ANR-12-INSE-0007) and MALTHY (ANR-13-INSE-0003), by the PGMO program of EDF and Fondation Mathématique Jacques Hadamard, and by the “Investissement d’avenir”, référence ANR-11-LABX-0056-LMH, LabEx LMH. M. Skomra was supported by a grant from Région Île-de-France.

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Allamigeon, X., Gaubert, S. & Skomra, M. Tropical Spectrahedra. Discrete Comput Geom 63, 507–548 (2020). https://doi.org/10.1007/s00454-020-00176-1

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