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.
UR - http://www.scopus.com/inward/record.url?scp=84891357778&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84891357778&partnerID=8YFLogxK
U2 - 10.1016/j.jnt.2013.09.014
DO - 10.1016/j.jnt.2013.09.014
M3 - Article
AN - SCOPUS:84891357778
SN - 0022-314X
VL - 137
SP - 93
EP - 121
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -