Newton Polyhedra, a New Formula for Mixed Volume, Product of Roots of a System of Equations

We generalize Vieta formula for the product of roots of a polynomial to the multidimensional case. We compute in the group (C∗)n the product of all roots of a system of n polynomial equations with sufficiently general Newton polyhedra. We present two different formulas for this product. In the first formula we use the so called Parshin symbols, in the second formula we use derivatives of the mixed volume with respect to vertices of all polyhedra. Both formulas employ certain combinatorial coefficients which characterize the relative location of Newton polyhedra at each vertex of their Minkowski sum. The technique of these coefficients is essential for our work. Using this technique we also prove a new formula for mixed volumes. To Vladimir Igorevich Arnold