Symmetric Kapranov and symmetric tropical ranks

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.