Higher order connectedness of amoebas

Background: Let $k$ be an algebraically closed non-archimedean field, and $\mathfrak{p}$ be a prime ideal in $R=k[x_1^{\pm1},\dots x_d^{\pm 1}]$. An argument in a forthcoming paper by Einsiedler, Lind, and Kapranov shows that the non-archimedean amoeba of $\mathfrak{p}$ is a connected set in $\mathbb{R}^d$. We also know that it is a homogeneous polyhedral complex of dimension $r$, where $r$ is the Krull dimension of $R/\mathfrak{p}$. But examples show that the amoeba may always have a higher type of connectivity as well.

Question: Let $\mathfrak{p}$ be a prime ideal in $R$, and $r$ be the Krull dimension of $R/\mathfrak{p}$. Form the finite graph whose vertices are the $r$-dimensional faces of the amoeba of $\mathfrak{p}$, and for which two vertices are joined if they share an $(r-1)$-face of the amoeba. Then is this graph always connected?

(contributed by Manfred Einsiedler, Doug Lind, and Rekha Thomas)

Back to the main index for Amoebas and tropical geometry.