On the Volume of Complex Amoebas

TL;DR

This paper investigates the geometric properties of amoebas of algebraic varieties in complex tori, establishing finiteness of their volume and providing bounds in specific cases, advancing understanding of their size and structure.

Contribution

It proves the finiteness of amoeba volumes for certain algebraic varieties and offers explicit estimates for rational curves, extending previous knowledge on amoeba geometry.

Findings

The area of complex algebraic curve amoebas is finite.

The volume of amoebas of $k$-dimensional varieties in $( ext{C}^*)^{n}$ is finite.

An estimate for the amoeba area of rational curves in terms of degree.

Abstract

The paper deals with amoebas of $k$-dimensional algebraic varieties in the algebraic complex torus of dimension $n\geq 2k$. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of $k$-dimensional algebraic variety in $(\mathbb{C}^*)^{n}$, with $n\geq 2k$, is finite.