Amoebas of half-dimensional varieties

TL;DR

This paper investigates the geometric properties of amoebas of half-dimensional algebraic varieties, providing bounds on their volume and the number of inverse points, advancing understanding of their structure in complex algebraic geometry.

Contribution

It establishes upper bounds for the volume of amoebas and the count of inverse image points for half-dimensional varieties, a novel contribution to the study of amoeba geometry.

Findings

Upper bounds for amoeba volumes are derived.

Bounds on the number of inverse points under amoeba and coamoeba maps are provided.

Results enhance understanding of the geometric complexity of algebraic varieties.

Abstract

An $n$-dimensional algebraic variety in $({\mathbb C}^\times)^{2n}$ covers its amoeba as well as its coamoeba generically finite-to-one. We provide an upper bound for the volume of these amoebas as well as for the number of points in the inverse images under the amoeba and coamoeba maps.