Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4595222 | Journal of Number Theory | 2007 | 8 Pages |
Abstract
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory