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.

Formulation

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

X₁, …, X_N,

and with coefficients in an algebraically closed field k (in fact, it suffices to assume k=C, the field of complex numbers). We consider these as a single vector-valued function

F: k^N → k^N

whose components are the F_i. The Jacobian determinant J of F is by definition the determinant of the N × N matrix consisting of the partial derivatives of F_i with respect to X_j:

$J=\det \left({\frac {\partial F_{i}}{\partial X_{j}}}\right).$

J is itself a function of the N variables X₁, …, X_N; indeed it is a polynomial function.

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.

Since k is algebraically closed and J is a polynomial, 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 G:k^N → k^N, then J is a non-zero constant.

The Jacobian conjecture is a strengthening of the converse: it states that

if J is a non-zero constant, then F has an inverse function G:k^N → k^N, and G is regular (in the sense that its components are given by polynomial expressions).

Results

The Jacobian conjecture has been proved for polynomials of degree 2, and it has also been shown that the general case 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.

It has been proved that the Jacobian conjecture is equivalent to the Dixmier conjecture. The Jacobian conjecture also implies the Ax-Grothendieck theorem.

See also

• Smale's problems
• Dixmier conjecture

References

• A. van den Essen (2001) [1994], "Jacobian conjecture", Encyclopedia of Mathematics, EMS Press
• 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

External links

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