Many systems arising in applications exhibit a sparse structure: not all monomials appear with a nonzero coefficient. We model the sparse structure of a polynomial by its Newton polytope, taking the convex hull of the exponents of the appearing monomials. The mixed volume of the tuple of Newton polytopes gives an upper bound for the number of isolated solutions of a polynomial system. For sparse systems the mixed volume is a much sharper bound than the classical Bezout number. In particular, for systems with randomly chosen coefficients, the mixed volume is an exact bound, and we say that mixed volumes count the number of isolated roots. Consequently, to solve sparse polynomials systems efficiently, we must use polyhedral homotopy continuation methods.
Johns Hopkins Random Talks Seminar, 11 April 2002.