**Conjecture**Let be a field of characteristic zero. A collection of polynomials in variables defines an automorphism of if and only if the Jacobian matrix is a nonzero constant.

The Jacobian determinant is the determinant of the matrix with . It is elementary to show that if the map 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.

## Bibliography

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