Abstract: As implied by Bezout's theorem, n generic polynomials of degrees d1,...,dn in C^n have exactly d1*d2*...*dk common roots. Here, the degrees of each polynomial are specified, but they are otherwise generic. Adding constraints, one may impose which *monomials* are involved in each polynomial, resulting in a 'sparse polynomial system'. The analogue of Bezout for sparse systems is the celebrated Bernstein-Kouchnirenko-Khovanskii (BKK) theorem.

The BKK theorem relates a solution count to the polyhedral geometry of the monomial support of a sparse system. Relating other algebro-geometric features to combinatorial data of sparse systems is an active area of research. I will give a survey of what is known about some of these connections, how the associated theorems are used in practice, and what has yet to be discovered.