= Hilbert–Samuel growth dimension
{c}
{title2=$d(R)=\deg H_R(t)$}
For a <Noetherian local ring> $(R,\mathfrak m)$, let $H_R(t)=\operatorname{length}(R/\mathfrak m^t)$. Its eventual <Hilbert-Samuel polynomial> has degree $d(R)$, equivalently the pole order at $1$ of the <Hilbert series> of $\operatorname{gr}_{\mathfrak m}R$. This is the local length-growth invariant, not the <embedding dimension>. The <prime-chain lower bound for local length> proves $\dim R\leq d(R)$; the full local dimension theorem gives equality.
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