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Klein polyhedra and lattices with positive norm minima

Oleg N. German
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Abstract

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 $\La$ have uniformly bounded determinants if and only if the facets and the edge stars of the vertices of the Klein polyhedron generated by $\La$ and related to the positive orthant have uniformly bounded determinants.

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Oleg N. German. 2006-06-04. Klein polyhedra and lattices with positive norm minima. https://arxiv.org/abs/math/0504483

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