---
tag: EGPP69XA4F
kind: statement
role: theorem
name: Finite determinacy theorem
msc: [58K40, 58K05]
uses: [7SWSRGDT4A, 1QE0Z0756D, 0YTCMFB4HC, 096Z9MP2ME, 0KZZ70ZHXP]
about: [7SWSRGDT4A]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Statement

Let $\mathcal{E}_n$ be the ring of [[germs|7SWSRGDT4A]] at $0$ of smooth functions $\R^n \to \R$.
Let $\mathfrak{m} \subset \mathcal{E}_n$ be its maximal [[ideal|096Z9MP2ME]], the germs that vanish at $0$.
For $f \in \mathcal{E}_n$, let $J(f)$ be the Jacobian ideal, generated by $\partial f / \partial x_1, \ldots, \partial f / \partial x_n$.
A germ $g$ is *right equivalent* to $f$ if $g = f \circ \varphi$ for a germ of [[diffeomorphism|0KZZ70ZHXP]] $\varphi \colon (\R^n, 0) \to (\R^n, 0)$.
Call $f$ *$k$-determined* if every $g \in \mathcal{E}_n$ with the same [[$k$-jet|1QE0Z0756D]] at $0$ as $f$ is right equivalent to $f$.

1. If $\mathfrak{m}^{k+1} \subseteq \mathfrak{m}^2 J(f)$, then $f$ is $k$-determined.
2. If $f$ is $k$-determined, then $\mathfrak{m}^{k+1} \subseteq \mathfrak{m} J(f)$.
3. Hence $f$ is $k$-determined for some $k$ if and only if $\mathfrak{m}^r \subseteq J(f)$ for some $r$.

## 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|0YTCMFB4HC]] 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|ERSD8D4HWX]].
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.

## Explanations

### L1 human

A smooth function near a point has a Taylor expansion with infinitely many terms.
Finite determinacy says that, for most interesting critical points, only finitely many terms matter.
After the first $k$ terms, any change of the higher terms can be absorbed by a change of coordinates.
For example, near a nondegenerate minimum, the quadratic terms already decide the shape completely.

### L2 human

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|M5437FP73G]] and the theory of [[versal unfoldings|KSAFWZVB8K]].