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On the Euclidean minimum of some real number fields. (English) Zbl 1161.11032

Let \(K\) be a number field; the Euclidean minimum is the infimum of the set of all real numbers \(\mu\) with the following property: for all \(x \in K\) there is a \(y \in O_K\) (its ring of integers) such that \(|N(x-y)| \leq \mu\). A number field \(K\) is Euclidean with respect to the norm if \(M(K) < 1\). For totally real number fields, a conjecture going back to Minkowski predicts that \(M(K) \leq 2^{-n} \sqrt{D_K}\), where \(n\) is the degree and \(D_K\) the discriminant of \(K\). This conjecture is known to hold for all \(n \leq 6\).
By studying ideal lattices in number fields and their invariants, the authors can show that \(M(K) \leq |D_K|\) for all number fields. For real quadratic number fields, they prove \(M(K) \leq \sqrt{D_K}/4\), and derive better bounds in special cases. For totally real cyclotomic fields of prime power conductor \(p^m\) they prove \(M(K) \leq 2^{-n} \sqrt{D_K}\), and they get stronger bounds if \(p > 2\). In the last section, they classify all totally real fields that are thin (a somewhat technical but natural property of a number field which implies that it is Euclidean).

MSC:

11R20 Other abelian and metabelian extensions
11H31 Lattice packing and covering (number-theoretic aspects)

Citations:

Zbl 1130.11066
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References:

[1] E. Bayer-Fluckiger, Lattices and number fields. Contemp. Math. 241 (1999), 69-84. · Zbl 0951.11016
[2] E. Bayer-Fluckiger, Ideal lattices. A panorama of number theory or the view from Baker’s garden (Zürich, 1999), 168-184, Cambridge Univ. Press, Cambridge, 2002. · Zbl 1043.11057
[3] E. Bayer-Fluckiger, Upper bounds for Euclidean minima. J. Number Theory (to appear). · Zbl 1130.11066
[4] J.W.S. Cassels, An introduction to the geometry of numbers. Springer Grundlehren 99 (1971). · Zbl 0209.34401
[5] J.H. Conway, N.J.A. Sloane, Low Dimensional Lattices VI: Voronoi Reduction of Three-Dimensional Lattices. Proc. Royal Soc. London, Series A 436 (1992), 55-68. · Zbl 0747.11027
[6] J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups. Springer Grundlehren 290 (1988). · Zbl 0634.52002
[7] P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers. North Holland (second edition, 1987) · Zbl 0611.10017
[8] The KANT Database of fields.
[9] F. Lemmermeyer, The Euclidean algorithm in algebraic number fields. Expo. Math. 13 (1995), 385-416. (updated version available via ). · Zbl 0843.11046
[10] C.T. McMullen, Minkowski’s conjecture, well-rounded lattices and topological dimension., Journal of the American Mathematical Society 18 (3) (2005), 711-734. · Zbl 1132.11034
[11] R. Quême, A computer algorithm for finding new euclidean number fields. J. Théorie de Nombres de Bordeaux 10 (1998), 33-48. · Zbl 0913.11056
[12] E. Weiss, Algebraic number theory. McGraw-Hill Book Company (1963). · Zbl 0115.03601
[13] M. Dutour, A. Schürmann, F. Vallentin, A Generalization of Voronoi’s Reduction Theory and Applications, (preprint 2005).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.