Jacobian conjecture

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.

For fixed N > 1 consider N polynomials F i , for 1 ≤ i ≤ N in the variables

: X 1, , X N ,

and with Coefficients in the complex numbers C . The Jacobian determinant J of the F i , considered as a vector-valued function (mathematics)

: F : C n → C n ,

is by definition the Determinant of the N × N matrix (mathematics) of the

: F ij ,

where F ij is the partial derivative of F i with respect to X j .

The condition

: J ≠ 0

enters into the inverse function theorem in multivariable calculus. In fact that condition for smooth functions (and so a fortiori for polynomials) ensures the existence of a local inverse function to F , at any point where it holds.

On the other hand in the polynomial case J is itself a polynomial. Since the complex numbers form an algebraically closed field J will be zero for some complex values of X 1, , X N , unless we have the condition

: J is a constant.

Therefore it is a relatively elementary fact that

:if F has an inverse function defined everywhere, then J is a constant.

The Jacobian conjecture is the converse: it states that

:if J is a non-zero constant function, then F has an inverse function.

A proof for the two-variable case was announced in 2004 by Carolyn Dean, and has been submitted for journal publication. Several sources have reported that her proof contains an error. A series of talks which she scheduled have been cancelled. See, for example, http://mathworld.wolfram.com/JacobianConjecture.html.

=External link=

*[http://www.math.purdue.edu/~ttm/jacobian.html Web page of T. T. Moh on the conjecture]