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

Algorithms for computing integral bases of an algebraic function eld 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 eld, to the case of a function eld. 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 diiculties 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 resultìntermediate expression swell' can be avoided. The Puiseux expansions generally introduce algebraic extensions. These extensions will not appear in the resulting integral basis. 1. Deenitions and notations The following conventions are used in the rest of the paper:-L is an algebraically closed eld of characteristic 0.-x is transcendental over L.-y is algebraic over L(x) with minimal polynomial f. We assume that y is integral over Lx]. This means that f is not only a monic polynomial over L(x) but over Lx]. The case where y is not integral over Lx] can be reduced to this case after multiplying y by an element of Lx].-K is a subbeld of L that contains all coeecients of f.-K is the algebraic closure of K in L.-n is the degree of f.-Lx] is the integral closure of Lx] in the algebraic function eld L(x; y).