Recognition problems

Background: Let $k$ be an algebraically closed non-archimedean field, and $\mathfrak{p}$ be a prime ideal in $k[x_1^{\pm1},\dots x_d^{\pm 1}]$. We know that the non-archimedean amoeba of $\mathfrak{p}$ coincides with the Bieri-Groves set of the algebra $A=k[x_1^{\pm1},\dots x_d^{\pm 1}]/\mathfrak{p}$, and is thus a homogeneous polyhedral complex whose dimension is the Krull dimension of $A$, and which is rationally defined over the value group of $k$. It also has the geometric property of total concavity, a sort of harmonic condition of spreading for the complex.

Question: Given a homogeneous polyhedral complex that is rationally defined over a dense subgroup of the reals and is also totally concave, what further conditions are necessary in order for it to be the amoeba of a prime ideal in the ring of Laurent polynomials over an algebraically closed non-archimedean field?

(contributed by Manfred Einsiedler and Doug Lind)

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