Rational functions over finite fields having continued fraction expansions with linear partial quotients Original Research Article

Let image be a finite field with q elements and let g be a polynomial in image with positive degree less than or equal to q/2. We prove that there exists a polynomial image, coprime to g and of degree less than g, such that all of the partial quotients in the continued fraction of g/f have degree 1. This result, bounding the size of the partial quotients, is related to a function field equivalent of Zarembaʹs conjecture and improves on a result of Blackburn [S.R. Blackburn, Orthogonal sequences of polynomials over arbitrary fields, J. Number Theory 6 (1998) 99–111]. If we further require g to be irreducible then we can loosen the degree restriction on g to deg(g)less-than-or-equals, slantq.