Maybe Jacobian Conjecture Counterexamples Are Not Rare

Maybe Jacobian Conjecture Counterexamples are not Rare – Win Vector LLC

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Maybe Jacobian Conjecture Counterexamples are not Rare

By John Mount on July 23, 2026 • ( Leave a comment )

The (now falsified) Jacobian conjecture was a well known conjecture expressing the hope that a family of functions had nice properties. In July 2026 Levent Alpöge exhibited a simple counter-example that he said was found by Claude Fable 5. Fields medalist Terence Tao wrote on this result try to relate the counterexample to known ideas and constructions. The issue being it seems hard to find such a counter-example by mere brute-force search, so one would like to find principles leading to the counter-example.

The (now disproven statement) was: when F is a polynomial map from Cn to Cn (n-vectors of complex numbers) such that the determinant of the Jacobian of F (write this as |D F|) is a non-zero constant independent location then: F has an inverse G that is also a polynomial map from Cn to Cn. The "determinant of the Jacobian" being a good stand-in for simple derivatives for vector maps.

Now if F and G are polynomial maps from Cn to Cn such that F(G(x)) = x everywhere then we would have |D F| and |D G| must be constant everywhere, as their product is 1 and they are both polynomials. We are using that the product of non-constant polynomials is never itself a constant. So the conjecture was checking if constant determinant of Jacobian implies a polynomial inverse, given we know a polynomial inverse implies a constant determinant of Jacobian. This is trying to see if an implication can be strengthened to an equivalence.

Many fields of mathematics study maps that are of the same type after composition. For instance the composition of two linear maps is again linear, and the composition of two neural nets is again a neural net. After composition one gets very interested in inverses: can one find inputs giving a specific output. The inverse of a linear map (when it exists) is again a linear map. However many neural nets cannot have inverses as the original neural net may map two different inputs to the same output. In this case finding an input that gives a specific output may be difficult.

Many polynomials do not have polynomial inverses. For example x2‘s inverse is +-sqrt(x), which is not single valued and also not a polynomial.

What is amazing is many non-trivial (not linear) polynomial maps do have polynomial inverses. For example consider the map:

F((z1, z2, z3)) = ( z1 - 2*z32, 12*z12 + z2 + 8*z3, 12*z1*z32 + z2/4 - 12*z34

It turns out this map has a polynomial map inverse:

G((y1, y2, y3)) = ( -(y14 - 20*y13*y3 - 50*y12*y2 + 150*y12*y32 + 500*y1*y2*y3 - 500*y1*y33 + 625*y22 - 1250*y2*y32 + 625*y34 - 625*y3)/125, -(y12 - 10*y1*y3 - 25*y2 + 25*y32)/25, -(y1 - 5*y3)/5

It is an amazing fact that F(G(x)) = x and G(F(x)) = x for all x in Cn.

How did we find F() and G()? We just performed brute force search using Python/sympy for the first two polynomials of F() and solved a symbolic system to get the third polynomial of F() and then all of G().

Now back to Alpöge's counter-example, here re-factored to have determinant Jacobian 1.

H((z1, z2, z3)) = ( 3*z13*z22*z3 + 9*z12*z23 + 6*z12*z2*z3 + 12*z1*z22 + 3*z1*z3 + z2, z13*z23*z3 + 3*z12*z24 + 3*z12*z22*z3 + 7*z1*z23 + 3*z1*z2*z3 + 4*z22 + z3, -z13*z3/2 - 3*z12*z2/2 + z1

This differers from the previous examples as H() is not 1 to 1, and therefore not invertible. For example H((-3, 1/3, 1)) = H((1, -2/3, -8)), showing H() is not 1 to 1, and loses information.

Could a simple longer Python/sympy search have found a counter-example, without all of the deep concepts Tao introduces? I think so. A heuristic argument says that counter-examples should not be too rare among high-degree functions with determinant Jacobian always 1. This is because composing any determinant Jacobian 1 function with a counter-example gives yet another counter-example. This should make it hard for counter-examples to be rare as we can map non-counter examples to counter-examples (assuming such composition doesn't encounter too many coincidences). So if we had a way to efficiently sample from higher degree maps with determinant and Jacobian 1, I expect we would find counter-examples quickly. The blockers being: the existence of any counter examples (now known thanks to Alpöge), and an efficient sampling method (not currently known to me).

This is a common situation in mathematics: it can be the case that most examples have a given property- but we have trouble showing this due to difficulties in realizing a sampling procedure.

Maybe Jacobian Conjecture Counterexamples Are Not Rare

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