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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
MSC:
 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