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For an $\mathfrak{m}$-primary ideal $I$ in a Noetherian local ring, the functionagrees for all sufficiently large $n$ with a polynomial in $n$. This is the Hilbert-Samuel polynomial, also called here the characteristic polynomial of $I$. Using $R/I^{n+1}$ instead merely shifts its variable.

In the Noetherian local ring $A=k[[u,v]]$ with maximal ideal $\mathfrak{m}=(u,v)$, the embedding dimension is two but ${\mathfrak{m}}^r$ needs $r+1$ generators for every $r\ge 1$. The degree-$r$ monomials $u^r,u^{r-1}v,\dots ,v^r$ generate ${\mathfrak{m}}^r$, and their images form a basis of ${\mathfrak{m}}^r/{\mathfrak{m}}^{r+1}$, so no smaller generating list exists. These ideals are $\mathfrak{m}$-primary: if $ab\in {\mathfrak{m}}^r$ and $a\notin {\mathfrak{m}}^r$, then $b$ cannot be a unit; hence $b\in \mathfrak{m}$ and $b^r\in {\mathfrak{m}}^r$. Consequently embedding dimension bounds the minimum over all maximal-primary ideals, because the maximal ideal itself is eligible, but does not bound the number of generators of each such ideal.

A prime chain of length $s$ in a Noetherian local ring forces its maximal-ideal length function to be at least $c{t}^s$ for large $t$. Quotient by the initial prime to reduce to a domain, choose a nonzero $a$ in the next prime, and use the Artin-Rees lemma to obtain $H_R(t)\ge H_R(t-c_0)+H_{R/(a)}(t)$. Induction on chain length and summing about $t/(2c_0)$ terms prove the bound. This yields $\dim R\le d(R)$ without assuming the Krull height theorem.

In a Noetherian local ring $(R,\mathfrak{m})$ of dimension $d$, a list $x_1,\dots ,x_d\in \mathfrak{m}$ whose generated ideal has radical $\mathfrak{m}$. Such lists exist, by successively avoiding the finitely many top-dimensional minimal primes. No list shorter than $d$ can generate a maximal-primary ideal, by the Krull height theorem.