×

Upper bounds for Euclidean minima of algebraic number fields. (English) Zbl 1130.11066

Let \(K\) be an algebraic number field with ring of integers \({\mathcal O}_K\), and let \(N\) denote the absolute value of the norm of elements. If for every \(x \in K\) there is a \(c\in {\mathcal O}_K\) such that \(N(x-c) < 1\), then \(K\) is said to be Euclidean (with respect to the norm). Thus we are led to study the real number \[ M(K) = \sup_{x \in K} \inf_{c \in {\mathcal O}_K} N(x-c) \] called the Euclidean minimum of \(K\). Clearly \(K\) is Euclidean if \(M(K) < 1\). In this article, it is shown that for a number field of degree \(n\) and with discriminant \(d_K\), we always have \(M(K) \leq 2^{-n}d_K\). This is a modest step towards Minkowski’s conjecture (now proved for all \(n \leq 6\)), which claims that \(M(K) \leq 2^{-n}\sqrt{d_K}\) for totally real fields. The author proves Minkowski’s conjecture for the maximal real subfields of the fields of \(p^r\)th roots of unity, and she also proves the bound \(M(K) \leq 2^{-n}\sqrt{| d_K| }\) for all fields of roots of unity. This is accomplished by the theory of ideal lattices in number fields and appealing to the theory of sphere packings.

MSC:

11R47 Other analytic theory
11R04 Algebraic numbers; rings of algebraic integers
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bachoc, C.; Batut, C., Etude algorithmique de réseaux construits avec la forme trace, Experiment. math., 1, 183-190, (1992) · Zbl 0787.11024
[2] Banaszczyk, W., New bounds in some transference theorems in the geometry of numbers, Math. ann., 296, 625-635, (1993) · Zbl 0786.11035
[3] Batut, C.; Quebbemann, H.-G.; Scharlau, R., Computations of cyclotomic lattices, Experiment. math., 4, 175-179, (1995) · Zbl 0873.11026
[4] Bayer-Fluckiger, E., Lattices and number fields, Contemp. math., 241, 69-84, (1999) · Zbl 0951.11016
[5] Bayer-Fluckiger, E., Cyclotomic modular lattices, J. théor. nombres Bordeaux, 12, 273-280, (2000) · Zbl 0997.11029
[6] Bayer-Fluckiger, E., Ideal lattices, (), 168-184 · Zbl 1043.11057
[7] Bayer-Fluckiger, E.; Martinet, J., Réseaux liés à des algèbres semi-simples, J. reine angew. math., 415, 51-69, (1994)
[8] E. Bayer-Fluckiger, G. Nebe, On the Euclidean minimum for some real number fields, J. Théor. Nombres Bordeaux, in press · Zbl 1161.11032
[9] Bayer-Fluckiger, E.; Suarez, I., Modular lattices over cyclotomic fields, J. number theory, 114, 394-411, (2005) · Zbl 1081.11047
[10] Cassels, J.W.S.; Cassels, J.W.S., The inhomogeneous minima of binary quadratic, ternary cubic, and quaternary quartic forms, Proc. Cambridge philos. soc., Proc. Cambridge philos. soc., 49, 519-520, (1952), (Addendum). Zbl 46.27662, Corr.: F.J. van der Linden, 1983 · Zbl 0046.27602
[11] J.-P. Cerri, Euclidean minima of totally real number fields. Algorithmic determination, Math. Comp., in press · Zbl 1209.11110
[12] J.-P. Cerri, Inhomogeneous and Euclidean spectra of number fields of unit rank greater than 1, J. Reine Angew. Math., in press · Zbl 1102.11057
[13] Clarke, L.E., Quart. J. math. (Oxford), 2, 308-315, (1951)
[14] Cohen, H., Advanced topics in computational number theory, Grad. texts in math., vol. 193, (2000), Springer · Zbl 0977.11056
[15] Conway, J.H.; Sloane, N.J.A., Sphere packings, lattices and groups, (1988), Springer · Zbl 0634.52002
[16] Craig, M., Extreme forms and cyclotomy, Mathematika, 25, 44-56, (1978) · Zbl 0395.10038
[17] Craig, M., A cyclotomic construction of Leech’s lattice, Mathematika, 25, 236-241, (1978) · Zbl 0413.10024
[18] Davenport, H., Linear forms associated with an algebraic number field, Quart. J. math. (Oxford), 2, 32-41, (1952) · Zbl 0047.27402
[19] Diaz y Diaz, F., Tables minorant la racine n-ième du discriminant d’un corps de degré n, (1980), Publications Mathématiques d’Orsay · Zbl 0482.12003
[20] Dyson, F.J., On the product of four non-homogeneous linear forms, Ann. of math., 49, 82-109, (1948) · Zbl 0031.15402
[21] Gruber, P.M.; Lekkerkerker, C.G., Geometry of numbers, (1987), North-Holland · Zbl 0611.10017
[22] Hardy, G.H.; Wright, E.M., An introduction to the theory of numbers, (1979), Oxford Univ. Press · Zbl 0423.10001
[23] Lemmermeyer, F., The Euclidean algorithm in algebraic number fields, Expo. math., 13, 385-416, (1995) · Zbl 0843.11046
[24] Lenstra, H.W., Euclid’s algorithm in cyclotomic fields, J. London math. soc. II, 10, 457-465, (1975) · Zbl 0313.12001
[25] Lenstra, H.W.; Lenstra, H.W.; Lenstra, H.W., Euclidean number fields, III, Math. intelligencer, Math. intelligencer, Math. intelligencer, 2, 99-103, (1979) · Zbl 0433.12004
[26] Lenstra, H.W., Euclidean ideal classes, (), 121-131 · Zbl 0401.12005
[27] Martinet, J., Petits discriminants des corps de nombres, (), 151-193 · Zbl 0491.12005
[28] Martinet, J., Réseaux parfaits des espaces euclidiens, (1996), Masson
[29] McMullen, C.T., Minkowski’s conjecture, well-rounded lattices and topological dimension, J. amer. math. soc., 18, 711-734, (2005) · Zbl 1132.11034
[30] Minkowski, H., Über die annährung an eine reelle größe durch rationale zahlen, Math. ann., 54, 91-124, (1901), (See also Ges. Abh. Chelsea Publ. Comp. XVI.) · JFM 31.0213.02
[31] Odlyzko, A., On conductors and discriminants, (), 377-407 · Zbl 0362.12006
[32] Remak, R.; Remak, R., Verallgemeinerung eines minkowskischen satzes, Math. Z., Math. Z., 18, 173-200, (1923) · JFM 49.0101.03
[33] R. Schoof, Computing Arakelov class groups, preprint · Zbl 1188.11076
[34] Skubenko, B.F., On Minkowski’s conjecture for \(n = 5\), Soviet math. dokl., Dokl. akad. nauk SSSR, 205, 1304-1305, (1972), Translation from: · Zbl 0263.10011
[35] van der Geer, G.; Schoof, R., Effectivity of Arakelov divisors and the theta divisor of a number field, Selecta math., 6, 377-398, (2000) · Zbl 1030.11063
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.