Georgia Institute of Technology College of Sciences

The classical Bernstein's theorem says that the number of roots of a system of sparse polynomials with generic coefficients equals the mixed volume of the Newton polytopes of the polynomials. We shall sketch a constructive proof by describing the solutions in the field of Puiseux series. The tropical Bernstein's theorem says that the number of tropical roots of a system of sparse tropical polynomials with generic coefficients equals the mixed volume of the Newton polytopes. We will prove this using the Huber--Sturmfels method for computing mixed volumes with regular mixed subdivisions of polytopes. Side topics: computation of mixed volumes, polyhedral homotopy continuation (finding complex solutions of a sparse polynomial system).