UK WILDCATS Seminar
**
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UK
WILDCATS
Seminar **

Work,
Interactions
&
Leading
Developments

in

Combinatorics,
Algebra,
Topology
&
Statistics

845 PATTERSON OFFICE TOWER

SPRING 2010

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"Higher integrality conditions and volumes of slices"

Fu Liu

University of California, Davis

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UK
WILDCATS
Seminar

Monday, March 29, 2010

4:00 pm, 845 Patterson Office Tower

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Abstract:

A polytope is integral if all of its vertices are lattice
points. The constant 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.