Jacobian conjecture

Uncommon card · No. 103136

In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n {\displaystyle n} -dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse. The case n = 2 {\displaystyle n=2} (two variables), also called the plane Jacobian conjecture or planar Jacobian conjecture, is the only case that remains unresolved as of 2026. The case n = 1 {\displaystyle n=1} is trivially true, since the derivative of a polynomial is a nonzero constant only if the degree of the polynomial is 1 , {\displaystyle 1,} and linear polynomial functions are invertible. On July 19, 2026, Levent Alpöge presented an explicit counterexample in three variables...

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