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

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 A1 + · · ·+Ar of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding C (A1 Ar). In this paper we extend this correspondence in a natural way to cover also noncoherent 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.