Hilbert polynomial

Summary

Let $M$ be a finitely generated graded module over a polynomial ring over a field. Its Hilbert polynomial gives the dimension of the degree $n$ part of $M$ for all large $n$. Its existence is the Hilbert–Serre theorem. For the coordinate ring of a projective variety, the degree of the Hilbert polynomial is the dimension of the variety. Its leading coefficient encodes the degree of the variety.