A Complexity Bound on Faces of the Hull Complex (English)

In: Discrete & Computational Geometry   ;  32 ,  4  ;  471-479  ;  2004
Springer-Verlag , New York; 2004

Abstract Given a monomial k[x1,. . . ,xn]-module M in the Laurent polynomial ring k[x1±1, . . . , xn±1], the hull complex is defined to be the set of bounded faces of the convex hull of the points {ta| xa ∈ M} for sufficiently large t. Bayer and Sturmfels conjectured that the faces of this polyhedron are of bounded complexity in the sense that every such face is affinely isomorphic to a subpolytope of the (n – 1)-dimensional permutohedron, which in particular would imply that these faces have at most n! vertices. In this paper we prove that the latter statement is true, and give a counterexample to the stronger conjecture.

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