Newton polyhedra, a new formula for mixed volume,
product of roots of a system of equations

A. G.  Khovanskii
		
		
  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.