GoKawiilThe notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.
Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse).
The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem, but is omitted here.) Also, from the fundamental theorem of algebra, once the Jacobian polynomial is non-zero, it must be constant. So the hypothesis “Jacobian is a non-zero constant” can be replaced with “ is locally invertible”. So the Jacobian conjecture can be viewed as an assertion that local invertibility implies global invertibility. The complex numbers can be easily replaced with other fields of characteristic zero by the Lefschetz principle, but I prefer to work in the concrete setting of the complex numbers.
Recently, it was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):
Theorem 2 (Counterexample to conjecture) There exists a polynomial which has non-zero constant Jacobian, but is not invertible.
The conjecture remains open in two dimensions, and is easy to establish in one dimension.
The example can be stated completely explicitly: one can take
and one can verify by a brief calculation that
and
While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomialhas degree seven, so a priori the Jacobianought to be a polynomial in three variables of degree as large as, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involvingcoefficients, which is much larger than thedegrees of freedom for a generic degree seven polynomial of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.
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