Symmetric Kapranov and symmetric tropical ranks

TL;DR

This paper investigates the tropical basis properties of minors of symmetric matrices and introduces two new symmetric matrix ranks derived from tropical geometry, revealing specific conditions under which these minors form a tropical basis.

Contribution

It establishes when the minors of symmetric matrices form a tropical basis and introduces symmetric tropical and Kapranov ranks for symmetric matrices, expanding tropical geometry tools.

Findings

Minors form a tropical basis for r=2, 3, n

Minors do not form a tropical basis for 4<r<n or r=4, n>12

Introduces symmetric tropical and Kapranov ranks for symmetric matrices

Abstract

This paper proves the $r \times r$ minors of an $n \times n$ symmetric matrix of indeterminates are a tropical basis when $r = 2$, $r = 3$, or $r = n$, and are not when $4 < r < n$ or $r = 4, n > 12$. In the process, it introduces two new notions of rank for symmetric matrices coming from tropical geometry, the symmetric tropical and the symmetric Kapranov rank, which are the symmetric versions of their standard counterparts defined by Develin, Santos, and Sturmfels.