Dimension computations for tropical determinantal varieties and prevarieties

TL;DR

This paper investigates the dimension differences between tropical prevarieties and tropical varieties related to minors of matrices, revealing cases where the prevariety's dimension exceeds that of the variety.

Contribution

It establishes conditions under which the tropical prevariety's dimension is larger than the tropical variety's, especially for minors of matrices and symmetric matrices.

Findings

Prevariety has greater dimension than the variety when minors are not a tropical basis.

The same dimension discrepancy occurs for symmetric matrices when r > 4.

Provides new insights into the structure of tropical determinantal varieties.

Abstract

This paper proves that when the $r \times r$ minors of an $m \times n$ matrix of indeterminates are not a tropical basis then the tropical prevariety has greater dimension than the tropical variety. It proves the same for the $r \times r$ minors of an $n \times n$ symmetric matrix of indeterminates when r > 4.