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

X₁, …, X_N,

and with coefficients in the complex numbers C. The Jacobian determinant J of the F_i, considered as a vector-valued function

F: Cⁿ → Cⁿ,

is by definition the determinant of the N × N matrix 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 X₁, …, 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

• Web page of T. T. Moh on the conjecture