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Inhomogeneous minima of mixed signature lattices. (English) Zbl 1419.11096

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 H. M. Davenport [Q. J. Math., Oxf. II. Ser. 3, 32–41 (1952; Zbl 0047.27402); Proc. Camb. Philos. Soc. 49, 190–193 (1953; Zbl 0051.03504)] and H. M. Davenport and H. P. F. Swinnerton-Dyer [Proc. Lond. Math. Soc., III. Ser. 5, 474–499 (1955; Zbl 0065.03201)]. 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 C. T. McMullen [J. Am. Math. Soc. 18, No. 3, 711–734 (2005; Zbl 1132.11034)] in the case of mixed signature in order to get explicit bounds for the Euclidean minimum.

MSC:

11H50 Minima of forms
11H06 Lattices and convex bodies (number-theoretic aspects)
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References:

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