Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller. It was widely publicized 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.

The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2016, there are no plausible claims to have proved it. Even the two variable case has resisted all efforts. There are no known compelling reasons for believing it to be true, and according to van den Essen (1997) there are some suspicions that the conjecture is in fact false for large numbers of variables. The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century.

Let N > 1 be a fixed integer and consider polynomials f₁, ..., f_N in variables X₁, ..., X_N with coefficients in a field k. Then we define a vector-valued function F: k^N → k^N by setting:

The Jacobian determinant of F, denoted by J_F, is defined as the determinant of the N × N Jacobian matrix consisting of the partial derivatives of f_i with respect to X_j:

then J_F is itself a polynomial function of the N variables X₁, ..., X_N.

It follows from the multivariable chain rule that if F has a polynomial inverse function G: k^N → k^N, then J_F has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

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