Far Eastern Mathematical Journal

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On Voronoi's cylindric minima theorem


A. V. Ustinov

2011, issue 2, . 213221


Abstract
Voronoi's algorithm for computing a system of fundamental units of a complex number field is based on a geometric properties of 3-dimensional lattices. This algorithm is based on Voronoi's theorem about cylindric minima for a lattice in general position. In the original proof and it's refinement published by Delone and Faddeev some significant cases were skipped. In the present we give a complete proof of Voronoi's theorem. The result is extended to arbitrary lattices.

Keywords:
lattice, Voronoi algorithm

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References

[1] G. Voronoi, Ob odnom obobshchenii algorifma nepreryvnykh drobei, Tipografiia Varshavskogo Uchebnogo Okruga, Varshava, 1896.
[2] B. N. Delone, D. K. Faddeev, Teoriia irratsional'nostei tret'ei stepeni, izd-vo AN SSSR, M.L., 1940.
[3] B. N. Delone, Peterburgskaia shkola teorii chisel, izd-vo AN SSSR, M.L., 1947.
[4] A. A. Illarionov, O tsilindricheskikh minimumakh trekhmernykh reshetok, Dal'nevost. matem. Zhurn., 11:1 (2011), 4855.

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