500 research outputs found
Rational Ehrhart quasi-polynomials
Ehrhart's famous theorem states that the number of integral points in a
rational polytope is a quasi-polynomial in the integral dilation factor. We
study the case of rational dilation factors and it turns out that the number of
integral points can still be written as a rational quasi-polynomial.
Furthermore the coefficients of this rational quasi-polynomial are piecewise
polynomial functions and related to each other by derivation.Comment: 15 pages, several changes in the expositio
Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes
Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational
polytope has a quasi-period less than the denominator of that polytope. This
phenomenon is poorly understood, and all known cases in which it occurs have
been proven with ad hoc methods. In this note, we present a conjectural
explanation for quasi-period collapse in rational polytopes. We show that this
explanation applies to some previous cases appearing in the literature. We also
exhibit examples of Ehrhart polynomials of rational polytopes that are not the
Ehrhart polynomials of any integral polytope.
Our approach depends on the invariance of the Ehrhart quasi-polynomial under
the action of affine unimodular transformations. Motivated by the similarity of
this idea to the scissors congruence problem, we explore the development of a
Dehn-like invariant for rational polytopes in the lattice setting.Comment: 8 pages, 3 figures, to appear in the proceedings of Integer points in
polyhedra, June 11 -- June 15, 2006, Snowbird, U
Ehrhart quasi-polynomials of almost integral polytopes
A lattice polytope translated by a rational vector is called an almost
integral polytope. In this paper we investigate Ehrhart quasi-polynomials of
almost integral polytopes. We study the relationship between the shape of the
polytopes and algebraic properties of the Ehrhart quasi-polynomials. In
particular, we prove that lattice zonotopes and centrally symmetric lattice
polytopes are characterized by Ehrhart quasi-polynomials of their rational
translations.Comment: ver 3: revisions on presentation
Periods of Ehrhart Coefficients of Rational Polytopes
Let P⊂R^n be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the kth dilate of P (k a positive integer) is a quasi-polynomial function of k — that is, a "polynomial" in which the coefficients are themselves periodic functions of k. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values
Generalized Ehrhart polynomials
Let be a polytope with rational vertices. A classical theorem of Ehrhart
states that the number of lattice points in the dilations is a
quasi-polynomial in . We generalize this theorem by allowing the vertices of
P(n) to be arbitrary rational functions in . In this case we prove that the
number of lattice points in P(n) is a quasi-polynomial for sufficiently
large. Our work was motivated by a conjecture of Ehrhart on the number of
solutions to parametrized linear Diophantine equations whose coefficients are
polynomials in , and we explain how these two problems are related.Comment: 18 pages, no figures; v2: Sections 4 and 5 added, proofs and
exposition have been expanded and clarifie
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