Klein polyhedra and lattices with positive norm minima

German, Oleg N.

Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 175-190.

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 $\Lambda$ 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.

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 ${ℝ}^n$. 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 $\Lambda$ 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 $\Lambda$ et correspondant à l’orthant positif, ont aussi leurs déterminants uniformément bornés.

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