The tropical analogue of the Helton-Nie conjecture is true (Q1994125)

The Helton-Nie conjecture suggests every convex semialgebraic set can be expressed as a projection of a spectrahedron. This conjecture was disproved by \textit{C. Scheiderer} [SIAM J. Appl. Algebra Geom. 2, No. 1, 26--44 (2018; Zbl 1391.90462)]. Scheiderer's proof utilizes the cone of nonnegative polynomials as a way to contradict the Helton-Nie conjecture. It is natural to ask for other counterexamples, possibly generated by more combinatorial techniques. This paper under review shows that over a real closed nonarchimedian field of Puiseux series the standard tropicalization using the valutions does not produce a counter-example to the Helton-Nie conjecture. The paper leaves it as an open problem to see if tropicalization with signs, in the spirit of Viro's patchworking, would produce a counter-example to the Helton-Nie conjecture.