Codimension of a singularity

Remarks

Example: for $f = x^4$ on $\R$, $J(f) = (x^3)$ and $\mathfrak{m} / J(f)$ has basis the classes of $x$ and $x^2$, so $\operatorname{codim}(f) = 2$. For a nondegenerate critical point $J(f) = \mathfrak{m}$, so the codimension is $0$. Non-example: the definition needs a critical point at $0$; for $f = x$ one has $J(f) = \mathcal{E}_1 \not\subseteq \mathfrak{m}$, and a regular point is not given a codimension in this sense. When it is finite, $\operatorname{codim}(f) = \dim_\R \mathcal{E}_n / J(f) - 1$. For a holomorphic germ, the same quotient of the ring of convergent power series has dimension the Milnor number $\mu(f)$. It equals the least number of parameters of a versal unfolding of $f$, not counting an additive constant. Conventions vary between authors; some count the additive constant as well and use $\dim_\R \mathcal{E}_n / J(f)$.