The tropical analogue of the Helton-Nie conjecture is true

TL;DR

This paper proves a tropical analogue of the Helton-Nie conjecture, showing that over certain nonarchimedean fields, convex semialgebraic sets and spectrahedron projections correspond via valuation, using game theory techniques.

Contribution

It establishes that in a tropical setting, convex semialgebraic sets and spectrahedral projections are equivalent through valuation, extending the understanding of convex algebraic geometry.

Findings

Convex semialgebraic sets and spectrahedral projections coincide under valuation in tropical setting.

The proof employs game theory methods to establish the main result.

The result provides a tropical analogue of a classical conjecture in real algebraic geometry.

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.