Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse. It was first conjectured in 1939 by Ott-Heinrich Keller, and widely publicized by Shreeram Abhyankar, as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.

The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors. As of 2018, it has not been proven, even for the two-variable case. Van den Essen provides evidence that the conjecture may be 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.