Geodesic
A closer look at lattice points in rational simplices
The electronic journal of combinatorics, Tome 6 (1999)
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We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an $n$-dimensional polytope with $n+1$ rational vertices, we use its description as the intersection of $n+1$ halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon.
DOI : 10.37236/1469
Classification : 11P21, 05A15, 52C07
Mots-clés : lattice point, \(n\)-dimensional polytopes, simplex, quasipolynomials 
@article{10_37236_1469,
     author = {Matthias Beck},
     title = {A closer look at lattice points in rational simplices},
     journal = {The electronic journal of combinatorics},
     year = {1999},
     volume = {6},
     doi = {10.37236/1469},
     zbl = {0934.11048},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1469/}
}
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Matthias Beck. A closer look at lattice points in rational simplices. The electronic journal of combinatorics, Tome 6 (1999). doi: 10.37236/1469

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