Euclidean minima of totally real number fields: Algorithmic determination

Abstract

This article deals with the determination of the Euclidean minimum M (K) of a totally real number field K of degree n ≥ 2, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree 2 to 8 and small discriminants, most of which were previously unknown. Tables are given at the end of this paper.

Norm-Euclidean Cyclic Fields of Prime Degree

“…In the case of = 3, we know that exactly 13 of the fields listed are norm-Euclidean, f = 73 being the only spurious value (see [21]). In the case of = 5, Godwin (see [8]) proved that f = 11 is norm-Euclidean and Cerri (see [4]) has verified this. Nothing seems to be known about the remaining fields in the table.…”

Some generalized Euclidean and 2-stage Euclidean number fields that are not norm-Euclidean

“…Let K be a number field of degree n. We have designed an algorithm which allows us to compute the Euclidean minimum of K, in particular when K is totally real [5], but also in the general case [3]. According to theoretical results [4], this algorithm can also give the upper part of the Euclidean spectrum of K and this yields new examples of number fields with interesting properties.…”

“…Without going into detail-these can be found in [5]-let us give nevertheless the theorem which justifies the algorithm and the main ideas that are behind it. Let us choose a constant k > 0 and a let us embed K into K ⊗ Q R =: K, which we can identify with R n , in which Z K is a lattice.…”

“…In this situation we know all the potential ξ ∈ K such that m K (ξ) > k, and since computing m K (ξ) is possible (again see [5] for more details), we know in fact that all of the ξ ∈ K such that m K (ξ) > k. To identify the elements ξ ∈ K such that m K (ξ) ≥ 1, it is sufficient to run the algorithm with k = 0.999, for instance.…”

Upper bounds for the Euclidean minima of abelian fields

“…The number field K is said to be Euclidean The study of Euclidean number fields and Euclidean minima is a classical one. However, little is known about the precise value of M (K) (see for instance [11] for a survey, and the tables of Cerri [7] for some numerical results). Hence, it is natural to look for upper bounds for M (K).…”