Klein polyhedra and lattices with positive norm minima
Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 175-190.

Un polyèdre de Klein est défini comme étant l’enveloppe convexe de tous les points non nuls d’un réseau qui se trouvent dans un orthant de l’espace n . On généralise à partir de ce concept la notion de la fraction continue. Dans cet article nous considérons les faces et les configurations étoilées, formées des arêtes issues de chaque sommet d’un polyèdre de Klein, comme les analogues multidimensionnels des quotient partiels, et nous définissons des charactéristiques quantitatives de ces “quotients partiels”, qui sont appelés “déterminants". Il est démontré que les faces de tous les 2 n polyèdres de Klein engendrés par un réseau Λ ont leurs déterminants uniformément bornés (dans leur ensemble) si, et seulement si, les faces et les étoiles d’arêtes des sommets du polyèdre de Klein, engendrées par Λ et correspondant à l’orthant positif, ont aussi leurs déterminants uniformément bornés.

A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of n . It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these “partial quotients”, so called determinants, are defined. It is proved that the facets of all the 2 n Klein polyhedra generated by a lattice Λ have uniformly bounded determinants if and only if the facets and the edge stars of the vertices of the Klein polyhedron generated by Λ and related to the positive orthant have uniformly bounded determinants.

DOI : 10.5802/jtnb.580
German, Oleg N. 1

1 Moscow State University Vorobiovy Gory, GSP–2 119992 Moscow, RUSSIA
@article{JTNB_2007__19_1_175_0,
     author = {German, Oleg N.},
     title = {Klein polyhedra and  lattices with positive norm minima},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {175--190},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     doi = {10.5802/jtnb.580},
     mrnumber = {2332060},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.580/}
}
TY  - JOUR
AU  - German, Oleg N.
TI  - Klein polyhedra and  lattices with positive norm minima
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2007
SP  - 175
EP  - 190
VL  - 19
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.580/
DO  - 10.5802/jtnb.580
LA  - en
ID  - JTNB_2007__19_1_175_0
ER  - 
%0 Journal Article
%A German, Oleg N.
%T Klein polyhedra and  lattices with positive norm minima
%J Journal de théorie des nombres de Bordeaux
%D 2007
%P 175-190
%V 19
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.580/
%R 10.5802/jtnb.580
%G en
%F JTNB_2007__19_1_175_0
German, Oleg N. Klein polyhedra and  lattices with positive norm minima. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 175-190. doi : 10.5802/jtnb.580. http://www.numdam.org/articles/10.5802/jtnb.580/

[1] O. N. German, Klein polyhedra and norm minima of lattices. Doklady Mathematics 406:3 (2006), 38–41. | MR

[2] P. Erdös, P. Gruber, J. Hammer, Lattice Points. Pitman Monographs and Surveys in Pure and Applied Mathematics 39. Longman Scientific & Technical, Harlow (1989). | MR | Zbl

[3] F. Klein, Uber eine geometrische Auffassung der gewohnlichen Kettenbruchentwichlung. Nachr. Ges. Wiss. Gottingen 3 (1895), 357–359.

[4] O. N. German, Sails and norm minima of lattices. Mat. Sb. 196:3 (2005), 31–60; English transl., Russian Acad. Sci. Sb. Math. 196:3 (2005), 337–367. | MR | Zbl

[5] J.–O. Moussafir, Convex hulls of integral points. Zapiski nauch. sem. POMI 256 (2000). | Zbl

[6] V. I. Arnold, Continued fractions. Moscow: Moscow Center of Continuous Mathematical Education (2002).

[7] V. I. Arnold, Preface. Amer. Math. Soc. Transl. 197:2 (1999), ix–xii.

[8] E. I. Korkina, Two–dimensional continued fractions. The simplest examples. Proc. Steklov Math. Inst. RAS 209 (1995), 143–166. | MR | Zbl

[9] T. Bonnesen, W. Fenchel, Theorie der konvexen Körper. Berlin: Springer (1934). | MR | Zbl

[10] B. Grünbaum, Convex polytopes. London, New York, Sydney: Interscience Publ. (1967). | MR | Zbl

[11] P. McMullen, G. C. Shephard, Convex polytopes and the upper bound conjecture. Cambridge (GB): Cambridge University Press (1971). | MR | Zbl

[12] G. Ewald, Combinatorial convexity and algebraic geometry. Sringer–Verlag New York, Inc. (1996). | MR | Zbl

[13] Z. I. Borevich, I. R. Shafarevich, Number theory. NY Academic Press (1966). | MR | Zbl

[14] J. W. S. Cassels, H. P. F. Swinnerton–Dyer, On the product of three homogeneous linear forms and indefinite ternary quadratic forms. Phil. Trans. Royal Soc. London A 248 (1955), 73–96. | MR | Zbl

[15] B. F. Skubenko, Minima of a decomposable cubic form of three variables. Zapiski nauch. sem. LOMI 168 (1988). | Zbl

[16] B. F. Skubenko, Minima of decomposable forms of degree n of n variables for n3. Zapiski nauch. sem. LOMI 183 (1990). | Zbl

[17] G. Lachaud, Voiles et Polyèdres de Klein. Act. Sci. Ind., Hermann (2002).

[18] L. Danzer, B. Grünbaum, V. Klee, Helly’s Theorem and its relatives. in Convexity (Proc. Symp. Pure Math. 7) 101–180, AMS, Providence, Rhode Island, 1963. | Zbl

Cité par Sources :