## Upper bounds for Euclidean minima of algebraic number fields.(English)Zbl 1130.11066

Let $$K$$ be an algebraic number field with ring of integers $${\mathcal O}_K$$, and let $$N$$ denote the absolute value of the norm of elements. If for every $$x \in K$$ there is a $$c\in {\mathcal O}_K$$ such that $$N(x-c) < 1$$, then $$K$$ is said to be Euclidean (with respect to the norm). Thus we are led to study the real number $M(K) = \sup_{x \in K} \inf_{c \in {\mathcal O}_K} N(x-c)$ called the Euclidean minimum of $$K$$. Clearly $$K$$ is Euclidean if $$M(K) < 1$$. In this article, it is shown that for a number field of degree $$n$$ and with discriminant $$d_K$$, we always have $$M(K) \leq 2^{-n}d_K$$. This is a modest step towards Minkowski’s conjecture (now proved for all $$n \leq 6$$), which claims that $$M(K) \leq 2^{-n}\sqrt{d_K}$$ for totally real fields. The author proves Minkowski’s conjecture for the maximal real subfields of the fields of $$p^r$$th roots of unity, and she also proves the bound $$M(K) \leq 2^{-n}\sqrt{| d_K| }$$ for all fields of roots of unity. This is accomplished by the theory of ideal lattices in number fields and appealing to the theory of sphere packings.

### MSC:

 11R47 Other analytic theory 11R04 Algebraic numbers; rings of algebraic integers
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