## You are here

HomeJacobian conjecture

## Primary tabs

# Jacobian conjecture

Let $F\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ be a polynomial map, i.e.,

$F(x_{1},\dots,x_{n})=(f_{1}(x_{1},\dots,x_{n}),\dots,f_{n}(x_{1},\dots,x_{n}))$ |

for certain polynomials $f_{i}\in\mathbb{C}[X_{1},\dots,X_{n}]$.

If $F$ is invertible, then its Jacobi determinant $\det(\partial f_{i}/\partial x_{j})$, which is a polynomial over $\mathbb{C}$, vanishes nowhere and hence must be a non-zero constant.

The *Jacobian conjecture* asserts the converse: every polynomial map
$\mathbb{C}^{n}\to\mathbb{C}^{n}$ whose Jacobi determinant is a non-zero constant
is invertible.

Synonym:

Keller's problem

Type of Math Object:

Conjecture

Major Section:

Reference

## Mathematics Subject Classification

14R15*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections