Klein polyhedra and lattices with positive norm minima

Oleg N. German[1]

  • [1] Moscow State University Vorobiovy Gory, GSP–2 119992 Moscow, RUSSIA

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 175-190
  • ISSN: 1246-7405

Abstract

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A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of ℝ 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.

How to cite

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German, Oleg N.. "Klein polyhedra and lattices with positive norm minima." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 175-190. <http://eudml.org/doc/249976>.

@article{German2007,
abstract = {A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of $\mathbb\{R\}^n$. 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 $\Lambda $ have uniformly bounded determinants if and only if the facets and the edge stars of the vertices of the Klein polyhedron generated by $\Lambda $ and related to the positive orthant have uniformly bounded determinants.},
affiliation = {Moscow State University Vorobiovy Gory, GSP–2 119992 Moscow, RUSSIA},
author = {German, Oleg N.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Klein polyhedra},
language = {eng},
number = {1},
pages = {175-190},
publisher = {Université Bordeaux 1},
title = {Klein polyhedra and lattices with positive norm minima},
url = {http://eudml.org/doc/249976},
volume = {19},
year = {2007},
}

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
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 175
EP - 190
AB - A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of $\mathbb{R}^n$. 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 $\Lambda $ have uniformly bounded determinants if and only if the facets and the edge stars of the vertices of the Klein polyhedron generated by $\Lambda $ and related to the positive orthant have uniformly bounded determinants.
LA - eng
KW - Klein polyhedra
UR - http://eudml.org/doc/249976
ER -

References

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  15. B. F. Skubenko, Minima of a decomposable cubic form of three variables. Zapiski nauch. sem. LOMI 168 (1988). Zbl0718.11026
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