Lattice points in lattice polytopes |
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Authors: | U Betke P McMullen |
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Institution: | (1) Mathematisches Institut, Universität Siegen, Hölderlinstrasse 3, D-5900 Siegen, Federal Republic of Germany;(2) Department of Mathematics, University College, Gower Street, WC1E 6BT London, U.K. |
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Abstract: | IfK is the underlying point-set of a simplicial complex of dimension at mostd whose vertices are lattice points, and ifG(K) is the number of lattice points inK, then the lattice point enumeratorG(K,t)=1+
n 1
G(nK)t
n
takes the formC(K, t)/(1–t)
d+1, for some polynomialC(K, t). Here,C(K, t) is expressed as a sum of local terms, one for each face ofK. WhenK is a polytope or its boundary, there result inequalities between the numbersG
r
(K), whereG(n K)=
r=0
d
n
r
G
r
(K). |
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Keywords: | |
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