Remarks on Euclidean minima

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 Cerriʼs 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 Cerriʼs 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.