Analytic varieties with finite volume amoebas are algebraic

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>=2k, we show that V is algebraic if and only if the volume of its amoeba is finite. Moreover, in this 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.