||Bezout's theorem tells us that two polynomials f(x,y) and g(x,y)
have (deg f)(deg g) common zeros in the projective plane. What about the
number of common zeros in the affine plane? Interestingly, this number is
related to the Newton polygons of f and g. We will begin with this first
glimpse of connection between convex geometry and algebraic geometry, and
proceed to introduce a recently developed generalization of this theory.