To prove that the Hilbert series (the generating function of the sequence of dimensions of homogeneous components) of a finitely generated commutative graded algebra is a rational function, the easiest way is to prove more general statement (Hilbert-Serre theorem) stating that the Hilbert series of a graded $k[x_1,\ldots,x_n]$-module is a rational function.
Vladimir Dotsenko
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