Analytic varieties with finite volume amoebas are algebraic

TL;DR

This paper proves that a generic analytic variety in a complex torus is algebraic if and only if its amoeba has finite volume, providing a new characterization linking algebraic and analytic geometry.

Contribution

It establishes a precise criterion connecting the finiteness of amoeba volume to the algebraicity of the variety, along with a comparison theorem for amoeba and coamoeba volumes.

Findings

Finite amoeba volume characterizes algebraic varieties when n ≥ 2k.

A comparison theorem relates amoeba and coamoeba volumes.

Applications include insights into k-linear spaces.

Abstract

In this paper, we study the amoeba volume of a given $k-$dimensional generic analytic variety $V$ of the complex algebraic torus $(\C^*)^n$. When $n\geq 2k$, we show that $V$ is algebraic if and only if the volume of its amoeba is finite. In this precise case, we establish a comparison theorem for the volume of the amoeba and the coamoeba. Examples and applications to the $k-$linear spaces will be given.