"Polyhedral Methods for Solving Polynomial Equations"

Bernd Sturmfels (New York University and UC Berkeley)

A basic problem in computational algebra is to find all zeros of a sparse system of polynomial equations. The situation is fairly well understood for complex zeros: their expected number is the mixed volume of the given Newton polytopes. This result due to Bernstein can be proved by an elementary algorithm using toric deformations.

Things are more difficult (and interesting) over the real numbers: Khovanskii has shown that the number of real roots is bounded by a function which is independent of the degree of the given equations. More precise upper bounds are stated in conjectures of Kouchnirenko and Itenberg-Roy. We discuss these results and conjectures, and we illustrate them with many colorful pictures of planar polygons.