Tropical Positivity and Determinantal Varieties

TL;DR

This paper introduces positive-tropical generators as positive analogues of tropical bases, applies them to determinantal varieties, and characterizes the positive parts of low-rank matrix tropicalizations, including a full classification for certain square matrices.

Contribution

It develops criteria for positive tropicalization of determinantal varieties and classifies the signed tropicalization for specific matrix cases, advancing tropical geometry understanding.

Findings

Criteria for positive tropicalization of determinantal varieties.

Full classification of signed tropicalization for certain square matrices.

Analysis of low-rank matrices of rank 2 and 3.

Abstract

We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.