In this paper we extend a theorem of Sternberg and Bileckii. We study a vector field, or a diffeomorphism, in the vicinity of a hyperbolic fixed point. We show that if the eigenvalues of the linear part (at the fixed point) satisfy 2N-algebraic conditions (where N > 1), then there is a CN-linearization in the vicinity of this fixed point. If the fixed point is stable, then the CN-linearization theorem follows when only (N + 1)-algebraic conditions are satisfied. Examples are given which show that the first of these results is sharp. An application to celestial mechanics is included.