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

✍ Scribed by Christian Friesen

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 119 KB
• Volume: 126
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.11.009

✦ Synopsis

Let F be a finite field with q elements and let g be a polynomial in F[X] with positive degree less than or equal to q/2. We prove that there exists a polynomial f ∈ F[X], 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 Zaremba's 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) q.