---
tag: NZASV9ZMGN
kind: definition
name: Versal unfolding
msc: [58K60, 58K35, 58K40]
defines: KSAFWZVB8K
uses: [7SWSRGDT4A, 0V33BNKZ9S]
sources:
  - url: https://en.wikipedia.org/wiki/Catastrophe_theory
---

## Definition

Let $f \colon (\R^n, 0) \to \R$ be a [[germ|7SWSRGDT4A]] of a [[smooth function|0V33BNKZ9S]].
An *$r$-parameter unfolding* of $f$ is a smooth germ $F \colon (\R^n \times \R^r, 0) \to \R$ with $F(x, 0) = f(x)$.
An $s$-parameter unfolding $G$ of $f$ is *induced* from $F$ if there are smooth germs $\Phi \colon (\R^n \times \R^s, 0) \to (\R^n, 0)$, $\alpha \colon (\R^s, 0) \to (\R^r, 0)$ and $c \colon (\R^s, 0) \to \R$ with $\Phi(x, 0) = x$ and
$$G(x, v) = F(\Phi(x, v), \alpha(v)) + c(v).$$
The unfolding $F$ is *versal* if every unfolding of $f$, with any number of parameters, is induced from $F$.
A versal unfolding with the least possible number $r$ of parameters is *universal*, or *miniversal*.

## Remarks

Let $\mathcal{E}_n$ be the ring of smooth germs at $0$ on $\R^n$ and $J(f)$ the ideal generated by the partial derivatives of $f$.
The versality theorem says that $F$ is versal if and only if
$$\mathcal{E}_n = J(f) + \R \cdot 1 + \R \cdot \frac{\partial F}{\partial u_1}(\cdot, 0) + \cdots + \R \cdot \frac{\partial F}{\partial u_r}(\cdot, 0),$$
where $u_1, \ldots, u_r$ are the parameters.
Example: for $f = x^3$, $J(f) = (x^2)$ and $\mathcal{E}_1 / J(f)$ has basis the classes of $1$ and $x$; for $F(x, u) = x^3 + u x$ one has $\partial F / \partial u (x, 0) = x$, so $F$ is versal, and universal, since one parameter is needed.
Non-example: $G(x, u) = x^3 + u x^2$ is an unfolding of $x^3$, but $\partial G / \partial u (x, 0) = x^2$ lies in $J(f)$, so the class of $x$ is not reached and $G$ is not versal.
Indeed, every function $G(\cdot, u)$ has a critical point near $0$, while $x^3 + u x$ has none for $u > 0$.
A germ has a versal unfolding with finitely many parameters if and only if $\mathcal{E}_n / J(f)$ is finite-dimensional.
If moreover $f$ has a critical point at $0$, a universal unfolding has $\dim_\R \mathcal{E}_n / J(f) - 1$ parameters.