The Lattice of Amoebas

TL;DR

This paper introduces a lattice-theoretic framework for amoebas of exponential sums, unifying existing theories and providing improved methods for analyzing their complement components.

Contribution

It develops a lattice structure on caissons, unifying lopsided amoebas and exponential sum amoebas, and offers enhanced certificates for amoeba complement components.

Findings

Lattice structure on caissons derived from subspace lattices.

Unified framework for lopsided and exponential amoebas.

Improved certificates for amoeba complement components.

Abstract

We study amoebas of exponential sums as functions of the support set $A$. To any amoeba, we associate a set of approximating sections of amoebas, which we call caissons. We show that a bounded modular lattice of subspaces of a certain vector space induces a lattice structure on the set of caissons. Our results unifies the theories of lopsided amoebas and amoebas of exponential sums. As an application, we show that our theory of caissons yields improved certificates for existence of certain components of the complement of an amoeba.