Remarks on Euclidean Minima

TL;DR

This paper extends previous results on Euclidean minima of number fields, proving that for fields of rank 3 or higher, the Euclidean minimum is isolated, attained, and computable in finite time, with complexity bounds for rank 2 or higher.

Contribution

It generalizes Cerri's results to higher rank fields using dynamical methods, establishing finiteness, attainability, and complexity bounds for Euclidean minima.

Findings

For rank ≥ 3, $M(K)$ is isolated and attained, and computable in finite time.

For rank ≥ 2, the complexity of computing $M(K)$ is bounded by degree, discriminant, and regulator.

The results apply to non-CM fields, extending previous work.

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 $K$.