**The Jacobian conjecture: **

*Suppose F: C^{N} → C^{N} is a complex polynomial map, F=(*

*F*

_{1}

_{,}

*F*

_{2}

_{,…,}

*F*

_{N}

*),*

*F*

_{i }

*are polynomials of the complex variables*

*X=(X*

_{1 , }

*X*

_{2}

*, …, X*

_{N}

*) . If the determinant of the derivative matrix DF(*

*X*

*) is a non-zero constant, then F must be one to one.*

In mathematics, the **Jacobian conjecture** is a celebrated problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller. It was later named and widely publicised by Shreeram Abhyankar, as an example of a question in the area of algebraic geometry that requires little beyond a knowledge of calculus to state. This is the * 16^{th}* problem in the list of the mathematical problems for the century proposed by Steve Smale in 2000.

The Jacobian conjecture has been proved for polynomials of degree 2 and can be reduced to the case of polynomials of degree 3 . In the case N=2 this conjecture has been verified for polynomials of degree less than 101, but is still unknown for polynomials of higher degrees. The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. See more in http://en.wikipedia.org/wiki/Jacobian_conjecture.

**References**

- O.H. Keller,
*Ganze Cremonatransformationen*Monatschr. Math. Phys. , 47 (1939) pp. 229–306 - A. van den Essen,
*Polynomial automorphisms and the Jacobian conjecture*, ISBN 3-7643-6350-9