The tropical analogue of the Helton-Nie conjecture is true
arXiv:1801.02089 · doi:10.1016/j.jsc.2018.06.017
Abstract
Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture. We show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods.
arXiv comment20 pages, 7 figures, this is an extended version of the preprint presented at MEGA 2017
References in corpus (10)
- Duality and separation theorems in idempotent semimodules
- Theta Bodies for Polynomial Ideals
- Tropical polyhedra are equivalent to mean payoff games
- Log-barrier interior point methods are not strongly polynomial
- Hyperfields for Tropical Geometry I. Hyperfields and dequantization
- Semidefinite Representation of the -Ellipse
- Tropical spectrahedra
- Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
- A Note on the Convex Hull of Finitely Many Projections of Spectrahedra
- The hyperring of adèle classes