The Amoeba Dimension of an Arbitrary Loopless Matroid

Let M be a loopless matroid on a finite set E with rank function r, and let Σ(M) ⊆ ℝ^E be the support of its matroid fan. Draisma, Eggleston, Pendavingh, Rau and Yuen defined adim(M) as the minimum of 2 dim(Σ(M) + R) − dim R over rational subspaces R ⊆ ℝ^E; for the matroid of a complex linear space this is the dimension of the amoeba of the space. They proved that adim(M) = min Σ_i (2r(P_i) − 1), the minimum over all partitions {P_1, …, P_k} of E, when M is realizable over ℂ, and asked whether this holds for every loopless matroid. The question appears in an Oberwolfach report, in a BIRS problem session and as Conjecture 1.4.1 of their paper. We show that the answer is yes, and that both numbers equal dim(Σ(M) + Σ(M)); the minimum is attained by a rational subspace in the braid arrangement. The lower bound dim(Σ(M) + Σ(M)) ≥ min Σ_i (2r(P_i) − 1) is a theorem of Bernstein, valid for all loopless matroids. The only new ingredient is the inequality 2 dim(Φ + R) − dim R ≥ dim(Φ + Φ) for every finite union of cones Φ and every subspace R, which follows from the Grassmann formula. Consequently S ↦ adim(M|S) is a matroid rank function, which answers one of the sub-questions of the BIRS problem. The others are answered by earlier results: the partition minimum for M|S is a matroid rank function of S (Draisma et al.), and by Bernstein's theorem it equals dim(Σ(M|S) + Σ(M|S)). The inequality can be strict for other fans. Exact computations on connected non-realizable matroids are consistent with the theorem. This is an unrefereed note.