The Cayley Trick, lifting subdivisions and the Bohne–Dress theorem on zonotopal tilings

Abstract

In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum ${\mathcal{A}}_1+\cdots +{\mathcal{A}}_r$ of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding $\mathcal{C}({\mathcal{A}}_{,}...,{\mathcal{A}}_r)$. In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne–Dress theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.