The amoeba dimension of a linear space

Abstract

Given a complex vector subspace $V$ of ${\mathbb{C}}^n$, the dimension of the amoeba of $V\cap ({\mathbb{C}}^{\ast }{)}^n$ depends only on the matroid that $V$ defines on the ground set $\{1,\dots ,n\}$. Here we prove that this dimension is given by the minimum of a certain function over all partitions of the ground set, as previously conjectured by Rau. We also prove that this formula can be evaluated in polynomial time.