Rational Ehrhart quasi-polynomials. (English) Zbl 1234.52010

Summary: 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. 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.


52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
Full Text: DOI arXiv


[1] Baldoni, V.; Berline, N.; Köppe, M.; Vergne, M., Intermediate sums on polyhedra: computation and real Ehrhart theory, (2010) · Zbl 1260.05006
[2] A. Barvinok, Computing the volume, counting integral points and exponential sums, in: Proceedings of the Eighth Annual Symposium on Computational Geometry, SCGʼ92, pp. 161-170.
[3] Barvinok, A., Computing the Ehrhart quasi-polynomial of a rational simplex, Math. comp., 75, 1449-1466, (2006) · Zbl 1093.52009
[4] Beck, M.; Haase, C.; Matthews, A.R., Dedekind-Carlitz polynomials as lattice-point enumerators in rational polyhedra, Math. ann., 314, 945-961, (2008) · Zbl 1144.11070
[5] Beck, M.; Robins, S., Computing the continuous discretely, (2007), Springer New York · Zbl 1114.52013
[6] Beck, M.; Sam, S.V.; Woods, K.M., Maximal periods of (Ehrhart) quasi-polynomials, J. combin. theory ser. A, 115, 517-525, (2008) · Zbl 1152.05006
[7] Brion, M.; Vergne, M., Residue forumulae, vector partition functions and lattice points in rational polytopes, J. amer. math. soc., 10, 797-833, (1997) · Zbl 0926.52016
[8] S. Chen, N. Li, S.V. Sam, Generalized Ehrhart polynomials, Trans. Amer. Math. Soc., in press, arxiv preprint at http://arxiv.org/pdf/1002.3658. · Zbl 1374.52017
[9] Ehrhart, E., Sur LES polyèdres rationnels homothétiques à n dimensions, C. R. acad. sci. Paris ser. A, 254, 616-618, (1962) · Zbl 0100.27601
[10] Haase, C.; McAllister, T.B., Quasi-period collapse and \(\mathit{GL}_n(\mathbb{Z})\)-scissors congruence in rational polytopes, (), 115-122 · Zbl 1163.52006
[11] McAllister, T.B.; Woods, K.M., The minimum period of the Ehrhart quasi-polynomial of a rational polytope, J. combin. theory ser. A, 109, 345-352, (2005) · Zbl 1063.52006
[12] McMullen, P., Lattice invariant valuations on rational polytopes, Arch. math., 31, 509-516, (1978/1979) · Zbl 0387.52007
[13] Sam, S.V.; Woods, K.M., A finite calculus approach to Ehrhart polynomials, Electron. J. combin., 17, (2010), Research Paper 68, 13 pp · Zbl 1197.52004
[14] Stanley, R.P., Two poset polytopes, Discrete comput. geom., 1, 9-23, (1986) · Zbl 0595.52008
[15] Woods, K.M., Computing the period of an Ehrhart quasi-polynomial, Electron. J. combin., 12, (2005), Research Paper 34, 12 pp · Zbl 1076.05005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.