TY - JOUR

T1 - Remarks on Euclidean minima

AU - Shapira, Uri

AU - Wang, Zhiren

N1 - Funding Information:
Uri Shapira is partially supported by the Advanced research Grant 228304 from the European Research Council . Zhiren Wang is partially supported by an AMS–Simons travel grant and the NSF Grant DMS-1201453 .

PY - 2014/4

Y1 - 2014/4

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jnt.2013.09.014

DO - 10.1016/j.jnt.2013.09.014

M3 - Article

AN - SCOPUS:84891357778

VL - 137

SP - 93

EP - 121

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -