Maximally Sparse Polynomials have Solid Amoebas

Let $f$ be an ordinary polynomial in $\mathbb{C}[z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n]$ with no negative exponents and with no factor of the form $z_1^{\alpha_1}… z_n^{\alpha_n}$where$\alpha_i$ are non zero natural integer. If we assume in addicting that $f$ is maximally sparse polynomial (that its support is equal to the set of vertices of its Newton polytope), then a complement component of the amoeba ${\mathcal{A}}_f$ in ${\mathbb{R}}^n$ of the algebraic hypersurface $V_f\subset ({\mathbb{C}}^{\ast }{)}^n$ defined by $f$, has order lying in the support of $f$, which means that ${\mathcal{A}}_f$ is solid. This gives an affirmative answer to Passare and Rullg\aa rd question in [PR2-01].