The linearization of an autonomous system in the neighborhood of a “node”-type singular point

In the article we study the connection between the smoothness class of the diffeomorphism, linearizing an autonomous system of class $C^\infty$ in the neighborhood of a “node”-type singular point, and the form of the resonance relations between the roots of the matrix of the linear part. We establish sufficient conditions for the existence of the linearizing diffeomorphism from a fixed smoothness class.