Inhomogeneous minima of mixed signature lattices

We establish an explicit upper bound for the Euclidean minimum of a number field which depends, in a precise manner, only on its discriminant and the number of real and complex embeddings. Such bounds were shown to exist by Davenport and Swinnerton-Dyer ([9–11]). In the case of totally real fields,...

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Bibliographic Details
Published in:Journal of number theory 2016-10, Vol.167, p.88-103
Main Authors: Bayer-Fluckiger, Eva, Borello, Martino, Jossen, Peter
Format: Article
Language:English
Subjects:
Euclidean minimum
Lattices
Mathematics
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Summary:We establish an explicit upper bound for the Euclidean minimum of a number field which depends, in a precise manner, only on its discriminant and the number of real and complex embeddings. Such bounds were shown to exist by Davenport and Swinnerton-Dyer ([9–11]). In the case of totally real fields, an optimal bound was conjectured by Minkowski and it is proved for fields of small degree. In this note we develop methods of McMullen ([20]) in the case of mixed signature in order to get explicit bounds for the Euclidean minimum.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2016.03.010