JournalsjemsVol. 2 , No. 2DOI 10.1007/s100970050003

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

  • Birkett Huber

    University of California, Berkeley, United States
  • Jörg Rambau

    Konrad-Zuse-Zentrum für Informationstechnik, Berlin, Germany
  • Domingo Gómez-Pérez

    Universidad de Cantabria, Santander, Spain
The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings cover

Abstract

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.