Amoeba finite basis does not exist in general

TL;DR

This paper proves that for most complex algebraic varieties of certain codimensions, their amoebas cannot be represented as finite intersections of hypersurface amoebas, and it refines the understanding of their boundary structures.

Contribution

It demonstrates the non-existence of finite amoeba bases for generic varieties of specific codimensions and provides a refined geometric characterization of hypersurface amoeba boundaries.

Findings

Amoebas of generic varieties with codimension greater than 1 do not have finite bases.

The paper refines the geometric understanding of the topological boundary of hypersurface amoebas.

It extends previous results by providing a more detailed boundary characterization.

Abstract

We show that the amoeba of a generic complex algebraic variety of codimension $1<r<n$ do not have a finite basis. In other words, it is not the intersection of finitely many hypersurface amoebas. Moreover we give a geometric characterization of the topological boundary of hypersurface amoebas refining an earlier result of F. Schroeter and T. de Wolff \cite{SW-13}.