An Algorithm for Computing an Integral Basis in an Algebraic Function Field

✍ Scribed by Mark van Hoeij

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 310 KB
• Volume: 18
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.1994.1051

✦ Synopsis

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 algorithm of Ford and Zassenhaus [Ford,1978], that computes the ring of integers in an algebraic number field, to the case of a function field.

It turns out that using algebraic geometry one can write a faster algorithm. The method we will give is based on Puiseux expansions. One can see this as a variant on the Coates' algorithm as it, is described in [Davenport,1981]. Some difficulties in computing with Puiseux expansions can be avoided using a sharp bound for the number of terms required which will be given in Section 3. In Section 5 we derive which denominator is needed in the integral basis. Using this result 'intermediate expression swell' can be avoided.

The Puiseux expansions generally introduce algebraic extensions. These extensions will not appear in the resulting integral basis.