Normal forms for differentiable maps near a fixed point

We propose in this paper a significant refinement of normal forms for differentiable maps near a fixed point. We give a method to obtain further reduction of classical normal forms with concrete and interesting applications. Our method leads to unique normal forms either with respect to general diffeomorphisms in certain cases or with respect to near identity diffeomorphisms in other cases. Our approach is rational in the sense that if the coefficients of a map are in a field K, so is its normal form. This revised version was published online in June 2006 with corrections to the Cover Date.