Bounds for the Euclidean minima of function fields (Q397887)

The Euclidean minimum is a numerical invariant which measures how elements of a number field can be approximated by algebraic integers. Its study is a classical topic in algebraic number theory, going back to Minkowski and Hurwitz. Many mathematicians have studied its properties; among others, we can cite Barnes, Swinnerton-Dyer, Cassels, Davenport and Fuchs.NEWLINENEWLINEThe Euclidean minima can also be defined in the function field case, replacing the absolute value by the degree. The existence of Euclidean function fields was studied by Armitage, Markanda, Madam, Queen and Smith but Euclidean minima never explicitly appear in their work.NEWLINENEWLINEIn this paper, the authors define Euclidean minima for function fields and give some bound for this invariant. They shown that the results are analogous to those obtained in the number field case.NEWLINENEWLINEContents: 1) Introduction 2) Geometric approach 3) The analogy with the number field case 4) The case of a general base field.