The amoeba dimension of a linear space

Abstract

Given a complex vector subspace $V$ of $\mathbb{C}^n$, the dimension of the amoeba of $V \cap (\mathbb{C}^*)^n$ depends only on the matroid that $V$ defines on the ground set $\{1,\ldots,n\}$. 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.