Membership problems

Background: For every ideal $\mathfrak{a}$ in $R_d=\mathbb{Z}[x_1^{\pm1},\dots x_d^{\pm 1}]$ there is a related dynamical system generated by $d$ commuting automorphisms of a compact abelian group via Pontryagin duality. For dynamics it is very important to determine when such systems have a finiteness condition called expansiveness. A theorem of Klaus Schmidt states in effect that when $\mathfrak{a}$ contains no nonzero integers, then the system is expansive if and only if the complex amoeba of $\mathfrak{a}$ does not contain the origin.

Question: Is there an algorithm to determine whether the complex amoeba of an ideal $\mathfrak{a}$ in $R_d$ contains the origin?

(contributed by Manfred Einsiedler and Doug Lind)

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