Finite determinacy theorem

Remarks

The proof of the first part uses the homotopy method: join $f$ to $g$ by $f_t = f + t(g - f)$ and look for diffeomorphisms $\varphi_t$ with $f_t \circ \varphi_t = f$. Differentiating in $t$ leads to an equation $g - f = -\sum_i \xi_i \, \partial f_t / \partial x_i$ for a time-dependent vector field $\xi$ vanishing at $0$, which can be solved by Nakayama's lemma because $g - f \in \mathfrak{m}^{k+1}$. Example: if $f$ has a nondegenerate critical point at $0$, then $J(f) = \mathfrak{m}$, so $\mathfrak{m}^3 = \mathfrak{m}^2 J(f)$ and $f$ is $2$-determined. With the normal form of a quadratic form this gives the Morse lemma. For $f = x^3$ on $\R$, $J(f) = (x^2)$ and $\mathfrak{m}^2 J(f) = (x^4) = \mathfrak{m}^4$, so $x^3$ is $3$-determined. For $f = x^2 y$ on $\R^2$, the ideal $J(f) = (xy, x^2)$ contains no power of $y$, so $f$ is not finitely determined. Mather and Tougeron proved such criteria; similar results hold for other equivalence relations and for holomorphic germs.

Going deeper

For a student

The theorem is what makes the classification of singularities a finite problem. A $k$-determined germ is fixed by a polynomial, its $k$-jet, so the classification takes place in a finite-dimensional jet space under the action of a finite-dimensional group. The condition $\mathfrak{m}^r \subseteq J(f)$ says that the local algebra $\mathcal{E}_n / J(f)$ is finite-dimensional; for holomorphic germs its dimension is the Milnor number. This underlies Arnold's classification of simple critical points and the theory of versal unfoldings.