Higher integrality conditions and volumes of slices

Fu Liu
University of California, Davis (UC Davis)
Department of Mathematics

A polytope is integral if all of its vertices are lattice points. The costant term of the Ehrhart polynomial of an integral polytope is known to be 1. I generalize this result by introducing the definition of k-integral polytopes, where 0-integral is equivalent to integral. I will show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees of less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.

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