AN ALGORITHM FOR COMPUTING AN INTEGRAL BASIS IN AN ALGEBRAIC FUNCTION-FIELD

Algorithms for computing integral bases of an algebraic function field are implemented in some computer algebra systems. They are used e.g. for the integration of algebraic functions. The method used by Maple 5 .2 and AXIOM is given by Trager in [Trager,1984]. He adapted an algori thm of Ford and Zassenhaus [Ford,1978], that computes the ring of inte gers in an algebraic number field, to the case of a function field. It turns out that using algebraic geometry one can write a faster algori thm. The method we will give is based on Puiseux expansions. One can s ee this as a variant on the Coates' algorithm as it is described in [D avenport,1981]. Some difficulties in computing with Puiseux expansions can be avoided using a sharp bound for the number of terms required w hich will be given in Section 3. In Section 5 we derive which denomina tor is needed in the integral basis. Using this result 'intermediate e xpression swell' can be avoided. The Puiseux expansions generally intr oduce algebraic extensions. These extensions will not appear in the re sulting integral basis.