Half-space behavior of amoebas

Background: Let $R_d=\mathbb{Z}[x_1^{\pm1},\dots x_d^{\pm 1}]$ and $\mathfrak{a}$ be an ideal in $R_d$ with $\mathfrak{a}\cap\mathbb{Z}=\{0\}$. The adelic amoeba of $\mathfrak{a}$ is the union of its complex amoeba and its $p$-adic amoebas over all rational primes $p$. If $\mathfrak{a}=\langle f\rangle$ is principal, an argument from dynamics shows that every 1-dimensional ray from the origin must intersect the adelic amoeba of $f$. There should be a version of this for general ideals, and it is enough to state this for prime ideals.

Question: Let $\mathfrak{p}$ be a prime ideal in $R_d$, and $r$ denote the Krull dimension of $R_d/\mathfrak{p}$. Then for every subspace of $\mathbb{R}^d$ with dimension $d-r+1$, does every half-space of the subspace intersect the adelic amoeba of $\mathfrak{p}$?

(contributed by Manfred Einsiedler and Doug Lind)

Back to the main index for Amoebas and tropical geometry.