Skip to main content Accessibility help
×
Home

Ehrhart Polynomials and Successive Minima

  • Martin Henk (a1), Achill Schürmann (a2) and Jörg M. Wills (a3)

Abstract

The Ehrhart polynomials for the class of 0-symmetric convex lattice polytopes in Euclidean n-space ℝn are investigated. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima of such polytopes are closely related by their geometric and arithmetic means. It is also shown that the roots of the Ehrhart polynomials of lattice n-polytopes with or without interior lattice points differ essentially. Furthermore, the structure of the roots in the planar case is studied. Here it turns out that their distribution reflects basic properties of lattice polygons.

Copyright

References

Hide All
1Bambah, R. P., Woods, A. C., and Zassenhaus, H., Three proofs of Minkowski's second inequality in the geometry of numbers. J. Austral. Math. Soc. 5 (1965), 453462.
2Barvinok, A., Computing the Ehrhart polynomial of a convex lattice polytope. Discrete Comput. Geom. 12 (1994), 3548.
3Barvinok, A., A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19 (1994), 769779.
4Barvinok, A. and Pommersheim, J. E., An algorithmic theory of lattice points in polyhedra. New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–1997, Math. Sci. Res. Inst. Publ., 38 (1994), 91147.
5Beck, M., De Loera, J., Develin, M., Pfeifle, J., and Stanley, R. P., Coefficients and roots of Ehrhart polynomials. Contemp. Math. 374 (2005), 1536.
6Beck, M. and Pixton, D., The Ehrhart polynomial of the Birkhoff polytope. Discrete Comput. Geom. 30 (2003), 623637.
7Beck, M. and Robins, S., Computing the continuous discretely: Integer-point enumeration in polyhedra. Springer (to appear), Preprint at http://math.sfsu.edu/beck/papers/ccd.html.
8Betke, U. and Gritzmann, P., An application of valuation theory to two problems of discrete geometry. Discrete Math. 58 (1986), 8185.
9Betke, U., Henk, M., and Wills, J. M., Successive-minima-type inequalities. Discrete Comput. Geom. 9 (1993), 165175.
10Betke, U. and Kneser, M., Zerlegungen und Bewertungen von Gitterpolytopen. J. Reine Angew. Math. 358 (1985), 202208.
11Betke, U. and McMullen, P., Lattice points in lattice polytopes. Monatsh. Math. 99 (1985), 253265.
12Bump, D., Choi, K.-K., Kurlberg, P., and Vaaler, J., A local Riemann hypothesis, I. Math. Z. 233 (2000), 119.
13Davenport, H., Minkowski's inequality for the minima associated with a convex body. Quarterly J. Math. 10 (1939), 119121.
14De Loera, J., Haws, D., Hemmecke, R., and Huggins, P., A user's guide for latte v1.1, software package latte, 2004, available at http://www.math.ucdavis.edu/~latte.
15De Loera, J., Hemmecke, R., Tauzer, J., and Yoshida, R., Effective lattice point counting in rational convex polytopes. J. Symb. Comput. 38 (2004), 12731302.
16Diaz, R. and Robins, S., The Ehrhart polynomial of a lattice polytope. Ann. Math. 145 (1997), 503518. Erratum in 146 (1997), 237.
17Ehrhart, E., Sur les polyedres rationnels homothétiques à n dimensions. C. R. Acad. Sci., Paris, Sér. A. 254 (1962), 616618.
18Ehrhart, E., Sur un problème de gèomètrie diophantienne linéaire. J. Reine Angew. Math. 227 (1967), 2549.
19Ehrhart, E., Sur la loi de rèciprocitè des polyèdres rationnels. C. R. Acad. Sci., Paris, Sèr. A 266 (1968), 695697.
20Gritzmann, P. and Wills, J. M., Lattice points. In Handbook of Convex Geometry (Gruber, P.M. and Wills, J.M., eds.), vol. B, North-Holland (Amsterdam, 1993).
21Gruber, P. M. and Lekkerkerker, C. G., Geometry of Numbers (2nd ed.), vol. 37, North-Holland Publishing Co. (Amsterdam, 1987).
22Grünbaum, B., Convex Polytopes (2nd ed. Prepared by Kaibel, V., Klee, V. and Ziegler, G. M.). Springer (New York, 2003).
23Henk, M., Inequalities between successive minima and intrinsic volumes of a convex body. Monatsh. Math. 110 (1990), 279282.
24Henk, M., Successive minima and lattice points. Rend. Circ. Mat. Palermo (2) Suppl., (2002), no. 70, part I, 377384.
25Hibi, T., Algebraic Combinatorics on Convex Polytopes. Carslaw Publications (Glebe, Australia, 1992).
26Hibi, T., Dual polytopes of rational convex polytopes. Combinatorica 12 (1992), 237240.
27Lagarias, J. C., Point lattices. Handbook of Combinatorics (Graham, R. L., Grötschel, M., and Lovász, L., eds.), vol. A, North-Holland (Amsterdam, 1995).
28Lagarias, J. C. and Ziegler, G. M., Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Canad. J. Math. 43 (1991), 10221035.
29Liu, F., Ehrhart polynomials of cyclic polytopes. J. Comb. Theory Ser. A 111 (2005), 111127.
30Liu, F., Ehrhart polynomials of lattice-face polytopes. http://arxiv.org/abs/math.co/0512616.
31Minkowski, H., Geometrie der Zahlen, Teubner (Leipzig-Berlin, 1896). (Reprinted: Johnson, New York, 1968.)
32Mordell, L. J., Lattice points in tetrahedron and generalized Dedekind sums. J. Indian Math. Soc. (N.S.) 15 (1951), 4146.
33Mustata, M. and Payne, S., Ehrhart polynomials and stringy Betti numbers. http://arxiv.org/abs/math.AG/0505054.
34Pick, G. A., Geometrisches zur Zahlenlehre. Sitzungsber. Lotus Prag 19 (1899), 311319.
35Pommersheim, J. E., Toric varieties, lattice points and Dedekind sums. Math. Ann. 295 (1993), 124.
36Reeve, J. E., On the volume of lattice polyhedra. Proc. London Math. Soc. 7 (1957), no. 3, 378395.
37Rodriguez-Villegas, F., On the zeros of certain polynomials. Proc. Amer. Math. Soc. 130 (2002), 22512254.
38Schneider, R., Convex bodies: The Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press (Cambridge, 1993).
39Scott, P. R., On convex lattice polygons. Bull. Austral. Math. Soc. 15 (1976), 395399.
40Siegel, C. L., Lectures on the Geometry of Numbers, Springer-Verlag (Berlin, 1989).
41Stanley, R. P., Decompositions of rational convex polytopes. Ann. Discrete Math. 6 (1980), 333342.
42Stanley, R. P., Enumerative combinatorics. vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press (Cambridge, 1997) (corrected reprint of the 1986 original).
43Weyl, H., On geometry of numbers. Proc. London Math. Soc. (2) 47 (1942), 268289.
44Wills, J. M., Kugellagerungen und Konvexgeometrie. Jahresber. Deutsch. Math.-Verein. 92 (1990), 2146.
45Wills, J. M., Minkowski's successive minima and the zeros of a convexity-function. Monatsch. Math. 109 (1990), 157164.
46Wills, J. M., On an analog to Minkowski's lattice point theorem. In The Geometric Vein: The Coxeter Festschrift (Davis, C., Grünbaum, B., and Sherk, F. A., eds.), Springer (New York, 1982), 285288.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed