Hilbert polynomials for the extension functor

Let R be a local ring, I⊆R an ideal, and M and N finite R-modules. In this paper we provide a number of results concerning the degree of the polynomial giving the lengths of the modules , when such a polynomial exists. Included among these results are a characterization of when this degree equals the Krull dimension of R, a characterization of when the degree of the polynomial associated to the first non-vanishing Ext under consideration equals the grade of I on M, and calculation of the degree of Hilbert polynomials associated to certain iterated expressions involving the extension functor.