Tropical 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 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.

arXiv commentv1: 23 pages, 3 figures; v2: 25 pages, 4 figures, stronger results in Section 4 + new example + minor revisions; v3: 31 pages, 4 figures, new theorem in Section 5 + more discussion in Sections 2 and 4

References in corpus (6)

• Log-barrier interior point methods are not strongly polynomial
• Tropicalizing the simplex algorithm
• Hyperfields for Tropical Geometry I. Hyperfields and dequantization
• Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
• Tropicalizing the positive semidefinite cone
• The tropical analogue of the Helton-Nie conjecture is true

Cited by in corpus (8)

• Log-barrier interior point methods are not strongly polynomial
• Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
• Tropical totally positive matrices
• The tropical analogue of the Helton-Nie conjecture is true
• Hyperbolic plane curves near the non-singular tropical limit
• Convergent Hahn Series and Tropical Geometry of Higher Rank
• Real tropicalization and analytification of semialgebraic sets
• Convex geometry over ordered hyperfields