Jacobian Conjecture

Importance: High ✭✭✭
Recomm. for undergrads: no
Posted by: Charles
on: July 6th, 2008
Conjecture   Let $ k $ be a field of characteristic zero. A collection $ f_1,\ldots,f_n $ of polynomials in variables $ x_1,\ldots,x_n $ defines an automorphism of $ k^n $ if and only if the Jacobian matrix is a nonzero constant.

The Jacobian determinant is the determinant of the matrix $ A $ with $ a_{ij}=\frac{\partial f_i}{\partial x_j} $. It is elementary to show that if the map $ F:k^n\to k^n $ is an automorphism, then the Jacobian determinant is a nonzero constant, by using the inverse map. The other direction has turned out to be rather difficult.

It is known that the Conjecture holds for polynomials of degree 2, and that the general case follows from a special case in degree 3.

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