26 research outputs found
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
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
Machine Learning the Dimension of a Polytope
We use machine learning to predict the dimension of a lattice polytope
directly from its Ehrhart series. This is highly effective, achieving almost
100% accuracy. We also use machine learning to recover the volume of a lattice
polytope from its Ehrhart series, and to recover the dimension, volume, and
quasi-period of a rational polytope from its Ehrhart series. In each case we
achieve very high accuracy, and we propose mathematical explanations for why
this should be so.Comment: 13 pages, 7 figure