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Choose homogeneous algebra generators $x_1,\dots ,x_r$ of positive degrees $d_1,\dots ,d_r$ for $S$; they exist because $S$ is Noetherian. For a finitely generated graded module $V={⨁}_j{V}_j$, its Poincare series of a graded module isThe Hilbert-Serre theorem states thatfor a Laurent polynomial $Q(t)\in \mathbb{Z}[t,t^{-1}]$.

We prove this by induction on $r$. For $r=0$, $S=k$ and $V$ is finite-dimensional, so its series is a Laurent polynomial. For $x=x_r$ of degree $d=d_r$, multiplication gives the exact sequence of graded modulesAdditivity of the Hilbert series yieldsBoth modules on the right are finitely generated over $k[x_1,\dots ,x_{r-1}]$, so the induction hypothesis proves the formula.

For the finitely generated commutative algebra $R$, let $R_j$ be the stated degree filtration. Its associated graded ringis a standard graded algebra generated by the initial forms of $x_1,\dots ,x_n$. SinceHilbert–Serre shows that the quantity ${\dim }_k{R}_j$ agrees with a polynomial for all sufficiently large $j$. The growth of a finitely generated commutative algebra has polynomial degree $\dim R$, independently of the chosen finite generating set.

Forwith the four displayed generators, a basis of $R_j$ consists of the Laurent monomials $Y_1^a{Y}_2^b$ satisfying $|a|+|b|\le j$. There are $4r$ points with $|a|+|b|=r>0$, and one point at the origin. Hence the growth of the two-variable Laurent polynomial algebra is