If we have a generic system of n polynomial equations in n variables, Bezout's theorem states that the number of solutions is the product of their degrees. However, sometimes polynomials aren't actually generic, and it's natural to ask if we can make Bezout's theorem more precise when there are restrictions on which polynomials can show up. The Bernstein-Kuchnirenko-Khovanskii (BKK) theorem provides a beautiful answer to one version of this question, showing that when we require our polynomials to be built out of specific monomials, the number of solutions relates to volumes of polytopes.

In this talk, we'll see a simple proof of a special case of the theorem using Hilbert polynomials. As time permits, we'll sketch out the general case, highlighting connections to toric varieties. This talk should be accessible to students in Math 631. Speaker(s): Will Dana