Maximally Sparse Polynomials have Solid Amoebas

TL;DR

This paper proves that maximally sparse polynomials have solid amoebas, meaning their amoeba complements correspond exactly to the vertices of the Newton polytope, confirming a conjecture by Passare and Rullgård.

Contribution

It establishes that maximally sparse polynomials always produce solid amoebas, confirming a key conjecture in the theory of algebraic hypersurfaces and amoebas.

Findings

Amoeba complements correspond to vertices of the Newton polytope.

Maximally sparse polynomials have solid amoebas.

Affirmative answer to Passare and Rullgård's question.

Abstract

Let $f$ be an ordinary polynomial in $\mathbb{C}[z_1,..., z_n]$ 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 $\mathscr{A}_f$ in $\mathbb{R}^n$ of the algebraic hypersurface $V_f\subset (\mathbb{C}^*)^n$ defined by $f$, has order lying in the support of $f$, which means that $\mathscr{A}_f$ is solid. This gives an affirmative answer to Passare and Rullg\aa rd question in [PR2-01].