Continued fractions in fields of positive characteristic

Abstract

Let Fq[X] be the ring of polynomials over Fq, the finite field of q elements, Fq(x) its field of quotients, Fq((1/x)) the completion of Fq(X) with respect to the infinite valuation, and Fq((X)) the completion of Fq(X) with respect to the X-adic valuation. This thesis deals with continued fractions in Fq((1/x)) and Fq((X)), which we shall refer to as function fields, and their characterization properties. There have been different kinds of continued fractions constructed over local fields, such as the p-adic number field; the two notable ones being due to Ruban and Schneider in the seventies. The Ruban type continued fraction, which mimics the classical continued fraction in the reals, was first developed in F2((1/x)) by Baum&Sweet, while the Schneider type continued fraction has never been seriously considered in function fields. Here we present the constructions of both types of continued fractions (Ruban and Schneider) in Fq((1/x)) and Fq((X)) and derive their basic properties. Next, it is shown that as in the classical case both continued fractions terminate if and only if they represent rational elements. As to the characterization of quadratic irrationals, it is well known that a real number is a quadratic irrational if and only if its classical continued fraction is periodic. In the function fields case, this result remains true for Ruban continued fraction, while for Schneider continued fraction, we can only show that a quadratic irrational belonging to a large class does indeed have periodic Schneider continued fraction. In the last part, we prove that should one try to construct continued fraction in function fields using the best approximation criteria, one will inevitably end up with Ruban continued fraction.