The Jacobian Conjecture

A one-line question about polynomials that resisted eighty-five years of attack — and the single, self-checking counterexample that appears to settle it. Notable less for the answer than for who found it, and how little we have to trust them.

The conjecture in plain terms

Take a polynomial map F that sends n complex variables to n complex outputs — each output an ordinary polynomial in the inputs. Its Jacobian is the matrix of first derivatives, and its Jacobian determinant is a single polynomial built from them. If F can be undone by another polynomial map, that determinant is forced to be a nonzero constant — the chain rule leaves no room for anything else.

In 1939 Ott-Heinrich Keller asked whether the converse holds: if the Jacobian determinant is a nonzero constant, must the map be invertible? It looks almost too clean to be hard. A nonzero constant determinant says the map is locally reversible everywhere, with no bad points where it folds or pinches. The Jacobian Conjecture is the claim that “locally reversible everywhere” forces “globally reversible” — that the map can never quietly wrap the space around and send two different points to the same place.

A useful reduction sharpens what a counterexample has to look like. Over the complex numbers a polynomial map with constant nonzero Jacobian that is merely injective is automatically bijective, with a polynomial inverse (Białynicki-Birula and Rosenlicht, 1962; also a corollary of the Ax–Grothendieck theorem). So the conjecture is equivalent to a bare statement about collisions: no such map is allowed to send two distinct points to the same image. To break it, you need exactly one collision.

Why it stayed open for eighty-five years

1. 1939 — Keller poses it. In Ganze Cremona-Transformationen Keller states the problem for integer coefficients in two variables. It generalizes to any field of characteristic zero and any dimension, and takes on a life far beyond its modest origin.
2. The graveyard of proofs. The conjecture became notorious for attracting proofs that collapse. Engel, Gröbner and others published arguments in the 1950s–60s that failed; van der Kulk and later surveyors catalogued the wreckage. Hyman Bass, Edwin Connell and David Wright opened their landmark 1982 survey by warning it had “earned a reputation as a graveyard.”
3. 1982 — Reduction to degree three. Bass, Connell and Wright (and independently Yagzhev) proved the astonishing fact that it is enough to settle the conjecture for cubic maps — even for maps of the special form x + (cubic). The general problem collapses onto a deceptively small target, which only deepened the mystery: even the cubic case would not yield.
4. 1994 — The real version is false. Sergey Pinchuk built a polynomial map of the plane over the real numbers with nowhere-zero Jacobian that is not injective. This is the trap that has swallowed intuition ever since: the real analogue is decisively false, yet Pinchuk's Jacobian is only nonvanishing, not constant, and the complex conjecture shrugged the counterexample off. The complex, constant-Jacobian statement stood.
5. 1998 — A place on Smale's list. Stephen Smale included the Jacobian Conjecture as Problem 16 in his list of mathematical problems for the twenty-first century, cementing its status alongside the Riemann Hypothesis and P vs NP as a problem the field expected to outlive most of its would-be solvers.
6. Deep connections, no resolution. The conjecture was shown to be entangled with the Dixmier Conjecture on the Weyl algebra (Tsuchimoto; Belov-Kanel and Kontsevich) and with questions in non-commutative geometry. Powerful company — and still, from 1939 to 2026, not a single accepted proof or counterexample in the complex case.

The counterexample

On 20 July 2026 the number theorist Levent Alpöge posted a map — crediting a question from Akhil Mathew and the model Fable for the search — that does the one forbidden thing. In coordinates (x, y, z), the map F : ℂ³ → ℂ³ is:

Two facts about it are all that the refutation needs:

• Its Jacobian determinant is exactly −2 — a nonzero constant, everywhere. So F satisfies the hypothesis of the conjecture precisely.
• It is not injective. The three distinct points (0, 0, −¼), (1, −3⁄2, 13⁄2) and (−1, 3⁄2, 13⁄2) all map to the single point (−¼, 0, 0).

A map that hits one point three times cannot be inverted. Constant nonzero Jacobian, yet not invertible: that is the conjecture, negated in a single object.

Why this one is different: it certifies itself

The history of hard problems is littered with claimed resolutions that needed years of expert refereeing — and often died in the attempt. When Vinay Deolalikar circulated a claimed proof that P ≠ NP in 2010, adjudicating it meant a hundred specialists reading a hundred pages for weeks. A proof is an argument, and arguments have to be trusted line by line.

A counterexample to the Jacobian Conjecture is the opposite kind of object. It is not an argument to be believed; it is a claim to be checked, and the check is mechanical and total. Differentiate three polynomials, take a 3×3 determinant, confirm it is the constant −2. Evaluate the map at three points, confirm they collide. Every step is exact arithmetic a computer algebra system finishes in milliseconds, with no floating-point doubt and no room for a gap. Here is the entire referee, in the language such a check is usually run in:

You do not have to trust Alpöge, or Mathew, or the model that found it. You do not even have to read the tweet carefully. You paste three polynomials into any algebra system and watch it return −2. The provenance of the object is irrelevant to its truth — which is exactly what you want when the discoverer is a machine.

What it means for the future of maths research

The lesson is not “an AI is smarter than mathematicians.” It is a statement about where the difficulty in this problem actually lived. For eighty-five years the obstacle was never verification — anyone since 1939 could have checked this map in an afternoon by hand. The obstacle was search: finding the needle in an unimaginably large haystack of candidate polynomials, most of which have a variable-dependent Jacobian and are discarded on sight. The conjecture sat in precisely the regime where finding is astronomically harder than checking.

That asymmetry — search hard, verification cheap — is the same gap that P vs NP is about, and it is the gap machines are now built to exploit. A model can propose and discard millions of structured candidates, steered by the reduction theory that tells it a counterexample can be taken cubic and low-dimensional. It has no need to be right; it needs to generate one object whose correctness a determinant will confirm. This suggests a concrete division of labour for the next era of the subject:

It is worth being clear about the limits. This touches only conjectures whose counterexamples are finite, checkable objects; it says nothing about proving the positive direction of anything, where the whole difficulty is quantifying over an infinity of cases. No determinant will ever prove the Riemann Hypothesis. What has changed is narrower and still large: for a whole class of open problems, being unable to find the counterexample was the only thing protecting the conjecture, and that protection is the part machines just removed.