Coefficient functions of the Ehrhart quasi-polynomials of rational polygons

Details Coefficient functions of the Ehrhart quasi-polynomials of rational polygons Coefficient functions of the Ehrhart quasi-polynomials of rational polygons

2009-06-03

arxiv.org/abs/0906.0674v1

Mathematics

Combinatorics

5 pages, 2 figures, 2008 International Conference on Information Theory and Statistical Learning (ITSL'08), held in Las Vagas,

Scientific paper

In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of \emph{non}-integral convex polygons. Define a \emph{pseudo-integral polygon}, or \emph{PIP}, to be a convex rational polygon whose Ehrhart quasi-polynomial is a polynomial. The numbers of lattice points on the interior and on the boundary of a PIP determine its Ehrhart polynomial. We show that, unlike the integral case, there exist PIPs with $b=1$ or $b=2$ boundary points and an arbitrary number $I \ge 1$ of interior points. However, the question of whether a PIP must satisfy Scott's inequality $b \le 2I + 7$ when $I \ge 1$ remains open. Turning to the case in which the Ehrhart quasi-polynomial has nontrivial quasi-period, we determine the possible minimal periods that the coefficient functions of the Ehrhart quasi-polynomial of a rational polygon may have.

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