Remarks on Euclidean Minima

Abstract

The Euclidean minimum $M(K)$ of a number field $K$ is an important numerical invariant that indicates whether $K$ is norm-Euclidean. When $K$ is a non-CM field of unit rank 2 or higher, Cerri showed $M(K)$, as the supremum in the Euclidean spectrum $Spec(K)$, is isolated and attained and can be computed in finite time. We extend Cerri's works by applying recent dynamical results of Lindenstrauss and Wang. In particular, the following facts are proved: (1) For any number field $K$ of unit rank 3 or higher, $M(K)$ is isolated and attained and Cerri's algorithm computes $M(K)$ in finite time. (2) If $K$ is a non-CM field of unit rank 2 or higher, then the computational complexity of $M(K)$ is bounded in terms of the degree, discriminant and regulator of $K$.