Jacobian conjecture

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

$$F(x_1,\mathrm{\dots },x_n)=(f_1(x_1,\mathrm{\dots },x_n),\mathrm{\dots },f_n(x_1,\mathrm{\dots },x_n))$$

for certain polynomials $f_i\in ℂ[X_1,\mathrm{\dots },X_n]$.

If $F$ is invertible, then its Jacobi determinant $det(\partial f_i/\partial x_j)$, which is a polynomial over $ℂ$, vanishes nowhere and hence must be a non-zero constant.

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

Title	Jacobian conjecture
Canonical name	JacobianConjecture
Date of creation	2013-03-22 13:23:46
Last modified on	2013-03-22 13:23:46
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 14R15
Synonym	Keller’s problem