Remarks on Euclidean minima

Uri Shapira, Zhiren Wang

Research output: Contribution to journalArticle

1 Citation (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)93-121
Number of pages29
JournalJournal of Number Theory
Volume137
DOIs
StatePublished - Apr 1 2014

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Euclidean
Number field
Unit
Euclidean norm
Supremum
Discriminant
Regulator
Computational Complexity
Invariant

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Shapira, Uri ; Wang, Zhiren. / Remarks on Euclidean minima. In: Journal of Number Theory. 2014 ; Vol. 137. pp. 93-121.
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Remarks on Euclidean minima. / Shapira, Uri; Wang, Zhiren.

In: Journal of Number Theory, Vol. 137, 01.04.2014, p. 93-121.

Research output: Contribution to journalArticle

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