On the Volume of Complex Amoebas

The paper deals with amoebas of $k$-dimensional algebraic varieties in the algebraic complex torus of dimension $n\ge 2k$. 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 $({\mathbb{C}}^{\ast }{)}^n$, with $n\ge 2k$, is finite.