Polynomial systems and mixed volumes

Bernstein's theorem (also known as the Bernstein-Khovanskii-Kushnirenko theorem) gives a bound on the number of nonzero solutions of a polynomial system of equations in terms of the mixed volume of its Newton polytopes. In this talk, we will give some background on polynomial systems and mixed volumes and then sketch a proof of Bernstein's theorem. We will also discuss some applications to enumerative combinatorics, including a bound on the number of doubly monic Laurent polynomials whose initial powers have vanishing constant term in terms of Eulerian numbers.