Real and Positive Tropicalizations of Symmetric Determinantal Varieties

TL;DR

This paper investigates the properties of real and positive tropicalizations of low-rank symmetric matrix varieties, revealing cases of coincidence and divergence between different tropicalization notions.

Contribution

It characterizes when real and complex tropicalizations coincide and compares two positive tropicalization definitions for symmetric matrices.

Findings

Real tropicalization matches complex tropicalization for rank two and corank one.

Positive tropicalizations by Speyer and Williams coincide for symmetric rank two matrices.

Positive tropicalizations differ for symmetric corank one matrices.

Abstract

We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We also show that the two notions of positive tropicalization introduced by Speyer and Williams coincide for symmetric rank two matrices, but they differ for symmetric corank one matrices.