Solid Amoebas of Maximally Sparse Polynomials

TL;DR

This paper proves Passare and Rullgard's conjecture that amoebas of maximally sparse Laurent polynomials are solid, with the complement's topology determined by the Newton polytope, using tropical degeneration analysis.

Contribution

It establishes the solidity of amoebas for maximally sparse polynomials and analyzes their stability and topological features through tropical degenerations.

Findings

Amoebas of maximally sparse polynomials are solid.

Stability of Ronkin function domains under tropical degenerations.

Classification of amoeba topology regimes based on polynomial support.

Abstract

The topology of amoebas of complex algebraic hypersurfaces is deeply connected to the combinatorics of the Newton polytope and the convex geometry of the Ronkin function. A long-standing conjecture of Passare and Rullgard asserts that the amoeba of a maximally sparse Laurent polynomial, whose support consists exactly of the vertices of its Newton polytope, must be solid, meaning that the complement of the amoeba has precisely as many connected components as the number of vertices of the Newton polytope. In this paper we prove this conjecture. The proof is based on a detailed analysis of the stability of the linearity domains of the Ronkin function under tropical degenerations of Laurent polynomials. We show that in the maximally sparse case no new slopes corresponding to interior lattice points can appear, forcing the amoeba complement to have the minimal possible topology. In addition, we establish stability results for the spines of degenerating amoebas, prove that the associated Newton subdivision stabilizes and coincides with the tropical subdivision for sufficiently small parameters, and derive geometric criteria controlling the appearance of lattice points in the dual subdivision. These results lead to a classification of three distinct regimes governing the topology of amoeba complements according to the position of the support relative to the Newton polytope.