Jacobian conjecture

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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.

The Jacobian conjecture has been proved for polynomials of degree 2, and it has also been shown that it follows from the special case where the polynomials are of degree 3.

The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. There are currently no plausible claims to have proved it.

[edit] See also

Smale's problems

[edit] References

A. van den Essen, "Jacobian conjecture" SpringerLink Encyclopaedia of Mathematics (2001)
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