Amoeba finite basis of solutions to a system of $n$ polynomials in $n$ variables

Abstract

We show that the amoeba of a complex algebraic variety defined as the solutions to a generic system of $n$ polynomials in $n$ variables has a finite basis. In other words, it is the intersection of finitely many hypersurface amoebas. Moreover, we give an upper bound of the size of the basis in terms of $n$ and the mixed volume $\mu$ of the Newton polytopes of the polynomial equations of the system. Also, we give an upper bound of the degree of the basis elements in terms of $\mu$.