Inhomogeneous minima of mixed signature lattices

TL;DR

This paper derives explicit upper bounds for the Euclidean minimum of mixed signature number fields based on their discriminant and embeddings, extending previous results and methods to more general cases.

Contribution

It develops new methods inspired by McMullen to establish explicit bounds for the Euclidean minimum in mixed signature fields.

Findings

Explicit upper bounds depending on discriminant and embeddings.

Extension of bounds to mixed signature fields.

Application of McMullen's methods to new cases.

Abstract

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. 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 in the case of mixed signature in order to get explicit bounds for the Euclidean minimum.