Tropical Spectrahedra
Copyright policy )Tropical Spectrahedra
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.
v1: 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
2010 Mathematics Subject Classification. 14T05, Other problems of combinatorial convexity, Determinants, permanents, traces, other special matrix functions, 90C22, 510, Combinatorial aspects of tropical varieties, Mathematics - Algebraic Geometry, Tropical Geometry, FOS: Mathematics, nonarchimedean fields, Mathematics - Combinatorics, Nonarchimedean fields, Semidefinite programming, 12J25, Semialgebraic sets and related spaces, Mathematics - Optimization and Control, Algebraic Geometry (math.AG), Vector spaces, linear dependence, rank, lineability, Arithmetic aspects of tropical varieties, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Semirings, spectrahedra, Spectrahedra, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], 14T05 (Primary), 90C22, 14P10, 12J25 (Secondary), Foundations of tropical geometry and relations with algebra, Non-Archimedean valued fields, tropical geometry, Optimization and Control (math.OC), 14P10, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Combinatorics (math.CO), valued fields, Valued fields
2010 Mathematics Subject Classification. 14T05, Other problems of combinatorial convexity, Determinants, permanents, traces, other special matrix functions, 90C22, 510, Combinatorial aspects of tropical varieties, Mathematics - Algebraic Geometry, Tropical Geometry, FOS: Mathematics, nonarchimedean fields, Mathematics - Combinatorics, Nonarchimedean fields, Semidefinite programming, 12J25, Semialgebraic sets and related spaces, Mathematics - Optimization and Control, Algebraic Geometry (math.AG), Vector spaces, linear dependence, rank, lineability, Arithmetic aspects of tropical varieties, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Semirings, spectrahedra, Spectrahedra, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], 14T05 (Primary), 90C22, 14P10, 12J25 (Secondary), Foundations of tropical geometry and relations with algebra, Non-Archimedean valued fields, tropical geometry, Optimization and Control (math.OC), 14P10, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Combinatorics (math.CO), valued fields, Valued fields
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