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 |1+...+|r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding �(|1,...,|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.
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Birkett Huber, Jörg Rambau, Domingo Gómez-Pérez, The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. 2 (2000), no. 2, pp. 179–198DOI 10.1007/S100970050003