TY - JOUR

T1 - Lattice point simplices

AU - Reznick, Bruce

N1 - Funding Information:
in part by the National Science Foundation and the Alfred P. Sloan Foundation

PY - 1986

Y1 - 1986

N2 - We consider simplices in Rmwith lattice point vertices, no other boundary lattice points and n interior lattice points, with an emphasis on the barycentric coordinates of the interior points. We completely classify such triangles under unimodular equivalence and enumerate. For example, in a lattice point triangle with exactly one interior point, that point must be the centroid. We discuss the literature for fundamental tetrahedra and prove that there are seven possible barycentric coordinates for a one-point tetrahedron. Following suggestions of P. Erdös, we prove that, for fixed m and n, there are only finitely many possible sets of barycentric coordinates for the interior points. We also discuss a generalization of Beatty's problem in combinatorial number theory which has arisen several times in recent years.

AB - We consider simplices in Rmwith lattice point vertices, no other boundary lattice points and n interior lattice points, with an emphasis on the barycentric coordinates of the interior points. We completely classify such triangles under unimodular equivalence and enumerate. For example, in a lattice point triangle with exactly one interior point, that point must be the centroid. We discuss the literature for fundamental tetrahedra and prove that there are seven possible barycentric coordinates for a one-point tetrahedron. Following suggestions of P. Erdös, we prove that, for fixed m and n, there are only finitely many possible sets of barycentric coordinates for the interior points. We also discuss a generalization of Beatty's problem in combinatorial number theory which has arisen several times in recent years.

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U2 - 10.1016/0012-365X(86)90015-4

DO - 10.1016/0012-365X(86)90015-4

M3 - Article

AN - SCOPUS:38249039625

SN - 0012-365X

VL - 60

SP - 219

EP - 242

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - C

ER -