### Abstract or Additional Information

The Newton diagram of a polynomial or analytic function is a powerful tool for studying its behaviour near a point. We introduce a "global version" of Newton diagram of a polynomial at a subvariety in order to study behaviour of the polynomial near generic points of the subvariety. We apply this notion to the "affine Bezout-problem" of counting number of isolated solutions (in ${\mathbb C}^n$) of a system of n polynomials and show that it is possible to arrive at the exact count by a recursive formula which involves at each step mixed volume of the faces of these Newton-type diagrams with respect to various (possibly singular) "flags of subvarieties". This in particular is a natural extension of the Bernstein-Kushnirenko-Khovanskii approach to the affine Bezout-problem.