Toric geometry and Grothendieck residues

We consider a system of $n$ algebraic equations $P_1=\dots=P_n=0$ in the torus $(\mathbb C\setminus 0)^n$. It is assumed that the Newton polyhedra of the equations are in a sufficiently general position with respect to one another. Let $\omega$ be any rational $n$-form which is regular on $(\mathbb C\setminus0)^n$ outside the hypersurface $P_1\dotsb P_n=0$. Formerly we have announced an explicit formula for the sum of the Grothendieck residues of the form $\omega$ at all roots of the system of equations. In the present paper this formula is proved.