---
tag: MGPPCDFQDG
kind: concept
name: Nondegenerate critical point
msc: [57R70, "58E05", 58K05]
sources:
  - url: https://en.wikipedia.org/wiki/Morse_theory
---

## Summary

A nondegenerate critical point of a smooth function is a [[critical point|0YTCMFB4HC]] at which the second derivative, the [[Hessian|0693Y2METT]], is a nondegenerate bilinear form.
Such points are isolated, and the [[Morse lemma|ERSD8D4HWX]] shows that near one of them the function is a sum of squares with signs.
A function whose critical points are all nondegenerate is a [[Morse function|KPQ96640ZJ]].

## Explanations

### L1 human

At a critical point the graph of a function is flat to first order.
The point is nondegenerate when the function still curves in every direction, up or down, to second order.
For example, $x^2 + y^2$ and $x^2 - y^2$ have nondegenerate critical points at the origin: a bowl and a saddle.
The function $x^3$ has a degenerate critical point at $0$, because its second derivative vanishes there and the graph is flat to second order.