Tropical spectrahedra

TL;DR

This paper introduces tropical spectrahedra as nonarchimedean valuations of spectrahedra over Puiseux series, providing a polyhedral characterization and linking semialgebraic sets to semilinear sets through valuation.

Contribution

It offers the first explicit polyhedral description of tropical spectrahedra and connects semialgebraic sets with semilinear sets via valuation, using Denef-Pas quantifier elimination.

Findings

Polyhedral characterization of generic tropical spectrahedra

Valuation maps semialgebraic sets to closed semilinear sets

Tropicalization of inequalities describes images of basic semialgebraic sets

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.