An arithmetic Bernstein-Kusnirenko inequality

The Bernstein-Kusnirenko inequality is a classic theorem in toric varieties which bounds the number of solutions of a system of polynomials in terms of the volume of their Newton polytopes. It shows how a geometric problem can be translated into a combinatorial, simpler one.

In this talk, I will present an arithmetic analogue to this result. This gives an upper bound on the complexity of the isolated zeros of a system in terms of integrals of concave functions.

This is a joint work with Martín Sombra (Universitat de Barcelona & ICREA).