Levent Alpöge published a counterexample to the Jacobian conjecture in three dimensions and thanked Fable for working on the problem
Levent Alpöge published a counterexample to the Jacobian conjecture in three dimensions and thanked Fable for working on the problem
On July 20, mathematician Levent Alpöge published the formula for a map of three complex variables. He thanked Akhil Mathew for posing the question and Fable for working on it. The Jacobian determinant of this map is always −2, but three distinct input points produce the same output.
The Jacobian conjecture, proposed by Keller in 1939, concerns polynomial maps, which are sets of formulas that assign another triple of complex numbers to each triple of complex numbers. For such a map, one can construct a table showing how each output changes when each input changes by a small amount. The number calculated from this table is called the Jacobian. The conjecture stated that if the Jacobian is a nonzero constant, the map must have a polynomial inverse.
The value −2 means that near any input point, a nearby input can be recovered uniquely from a nearby output. A single inverse formula valid throughout the entire space requires more: every output point must correspond to exactly one input point.
In his post, Alpöge gave three distinct input points: (0, 0, −1/4), (1, −3/2, 13/2), and (−1, 3/2, 13/2). The map sends all three to (−1/4, 0, 0). One output therefore has three distinct preimages, so the map has no global inverse formula.
A mathematical note reduces the search for preimages of an arbitrary output point to a cubic equation, which is an equation of degree three. For (−1/4, 0, 0), it factors as −s(s−2)(s+2)/2. Its roots, 0, 2, and −2, correspond exactly to the three published points.
This map has sequences of input points that tend to infinity while their outputs converge to a finite point. A constant Jacobian controls the map near each point, but not across the entire space.
In the original post, Alpöge described the participants' contributions as follows:
“The Jacobian conjecture is false. Thanks to my close friend Akhil for the question and my other close friend Fable for working on it during the World Cup final.”
The published materials allow the formula and its properties to be checked independently. They do not disclose who completed each step of the search.
A week earlier, physicist Yuji Tachikawa reported that Claude Fable had found a computational error in a stalled string theory problem and developed his approach further. In the present case, the publication provides a formula, a constant Jacobian, and three points that can each be checked independently.
Adding new coordinates to this map and leaving them unchanged extends the counterexample to any number of variables starting from three. The two-dimensional case of the Jacobian conjecture remains open.