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

SC 98-44 Birkett Huber, Jörg Rambau, Francisco Santos: 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 $\mathcal{C}(\mathcal{A}_1,\dots,\mathcal{A}_r)$.
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.
Keywords: Polyhedral subdivision, fiber polytope, mixed subdivision, lifting subdivision, Minkowski sum, Cayley Trick, Bohne-Dress Theorem
MSC: 52B11, 52B20, 14M25