= Growth of a finitely generated commutative algebra
Let a commutative $k$-algebra $R$ be generated by $x_1,\ldots,x_n$, and let $R_j$ be the span of words of total degree at most $j$ in those generators. The <associated graded ring> of this filtration is a <standard graded algebra>, and
$$
\dim_kR_j=\sum_{i=0}^j\dim_k(R_i/R_{i-1}).
$$
The <Hilbert-Serre theorem> therefore makes $\dim_kR_j$ eventually a polynomial in $j$. Its degree is $\dim R$; this follows by applying the Rees-ring deformation, whose special fibers are $R$ and the associated graded ring, so they have the same <Krull dimension>.
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