The Jacobian conjecture was devised by German mathematician Ott-Heinrich Keller in 1939.

It states :

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."

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Put more formally :

Let k have characteristic 0. If J_F is a non-zero constant, then F has an inverse function G: k^N → k^N which is regular, meaning its components are polynomials."

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Since 1939, there have been many published 'solutions' to the problem. To date however, every one has been rejected by the mathematical community due to errors of one sort or another.

The quotes are from the Wikipedia article on the conjecture.