The volume polynomial of lattice polygons

Let P and Q be lattice polygons in the plane. Then their normalized volumes V(P) and V(Q) and their normalized mixed volume V(P,Q) satisfy the classical Minkowski inequality V(P)V(Q)<= V(P,Q)^2. We show that for any triple of nonnegative integers a,b,c that satisfy the inequality ab<= c^2 there exist lattice polygons P and Q in R^2 such that V(P)=a, V(Q)=b, and V(P,Q)=c. This shows that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons, which solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

This is joint work with Ivan Soprunov.