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Au sujet de l’algorithme ”de Coates”. (Concerning the Coates algorithm). (French) Zbl 0602.14028
Let C be a projective algebraic curve defined over an algebraically closed field K of characteristic zero. Denote by K(C) the function field of C over K. Thus K(C) is a finite algebraic extension of a rational subfield K(x) of degree \(n=[K(C):K(x)]\). Coates’ algorithm is aimed at computing a basis of the K-vector space L(D) of multiples in K(C) of a divisor D on C.
The author discusses the history of the algorithm, gives a description of its basic two parts, indicates some applications of it and comments on problems concerning its implementation. She also announces an improved version of the algorithm working with \(n\times n\)-matrices over the field of definition \(K_ 0\) of C instead of K. In its first complete implementation by D. Maurer [”Der Algorithmus von Coates”, Diploma thesis (Saarbrücken 1982)] \(n\times n\)-matrices over \(K_ 0\) had to be used instead, where m is a positive integer such that \(n\leq m\leq n^ 2\). Since the first part of Coates’ algorithm in particular also yields an integral basis of K(C) with respect to the polynomial ring K[x], it would be interesting to compare it with an analogue of the Ford/Zassenhaus algorithm for determining an integral basis of an algebraic number field [cf. R. Böffgen, ”Der Algorithmus von Ford/Zassenhaus zur Berechnung von Ganzheitbasen in Polynomalgebren”, Ann. Univ. Sarav., Ser. Math. 1, No.3, 60-129 (1986)].
Reviewer: H.G.Zimmer
14H25 Arithmetic ground fields for curves
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
12-04 Software, source code, etc. for problems pertaining to field theory
14C20 Divisors, linear systems, invertible sheaves
11R58 Arithmetic theory of algebraic function fields