Segre Classes as Integrals over Polytopes

We express the Segre class of a monomial scheme—or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections— in terms of an integral computed over an associated body in euclidean space. The formula is in the spirit of the classical Bernstein-Kouchnirenko theorem computing intersection numbers of equivariant divisors in a torus in terms of mixed volumes, but deals with the more refined intersection-theoretic invariants given by Segre classes, and holds in the less restrictive context of ‘r.c. monomial schemes’.