Piecewise polynomial functions, convex polytopes and enumerative geometry

Abstract

0. Introduction. This paper explores some of the connections between the objects of its title. It is based on a new approach to McMullen’s polytope algebra, and on its relation with equivariant cohomology of toric varieties. In particular, we give another proof of a recent result of Fulton and Sturmfels, which identifies the polytope algebra with the direct limit of all Chow rings of smooth, complete torus embeddings (see [14]). On the other hand, we obtain a version of the classical theorem of Bezout, which holds in any spherical homogeneous space. This generalizes a theorem of Bernstein and Kouchnirenko: The number of common points to d hypersurfaces in general position in a d-dimensional torus is d! times the mixed volume of the associated Newton polytopes (see [2], [18] and also [13] 5.5). Given a finite-dimensional vector space V over an ordered field K, there is a wellknown correspondence between convex polytopes in the dual space V ∗ and piecewise linear convex functions on V . Namely, to any convex polytope, we associate its support function. Denote by R the algebra generated by the support functions of all convex polytopes, in the algebra of continuous functions on V . In the first section of this paper, we study the algebra R when K is the field of rational numbers. It turns out (see 1.5) that R is the algebra of continuous, piecewise polynomial functions on V ; in particular, R contains the algebra of polynomial functions. We prove that any choice of coordinate functions on V defines a regular sequence in R; moreover, the quotient of R by the ideal generated by V ∗ is isomorphic to the rational polytope algebra of McMullen (see 1.3, 1.5; our proof is based on work of Morelli, see [21]). This explains the non-trivial grading of the polytope algebra, by the obvious grading of R. More generally, the quotients of R by powers of the ideal generated by V ∗, are isomorphic to the higher versions of the polytope algebra, considered recently by McMullen, Pukhlikov and Khovanskii; see [20], [24]. In fact, we study the algebra R as the direct limit of its subalgebras RΣ consisting of functions which are piecewise polynomial with respect to a fixed fan Σ. To such a fan is