Georgia Institute of Technology College of Sciences

Bernstein's theorem is a classical result which computes the number of common zeros in (C*)^n of a generic set of n Laurent polynomials in n variables. The theorem of the Newton polygon is a ubiquitous tool in arithmetic geometry which calculates the valuations of the zeros of a polynomial (or convergent power series) over a non-Archimedean field, along with the number of zeros (counted with multiplicity) with each given valuation. We will explain in what sense both theorems are very special cases of a lifting theorem in tropical intersection theory. The proof of this lifting theorem builds on results of Osserman and Payne, and uses Berkovich analytic spaces and extended tropicalizations of toric varieties in a crucial way, as well as Raynaud's theory of formal models of analytic spaces. Most of this talk will be about joint work with Brian Osserman.