The Jacobian Conjecture

How a machine broke an 87-year-old problem, and why it counts.

At 2:19 in the morning on July 20th, 2026, a mathematician named Levent Alpöge posted a tweet with no capital letters in it.¹ Inside the tweet was a formula. Inside the formula was the end of a question that had been open since 1939.

The question is called the Jacobian Conjecture. It is one of those problems that sounds almost childish when you say it out loud, needs almost no machinery to state, and then refuses to die for a century. Stephen Smale put it on his list of the eighteen problems he thought would define mathematics in the 21st century. It sits at number sixteen, two places above the one that asks what the limits of intelligence are, artificial and human.² It has attracted so many wrong proofs that the wrong proofs have their own literature.

Alpöge found the counterexample with Claude Fable, a language model made by Anthropic, where he also works.³ He credits it in the tweet, between thanking a friend for suggesting the problem and noting that the machine had been working through the World Cup final.

I want to show you what fell. Not the formal statement, which will not mean anything to you yet, but the actual shape of the thing, built up from a picture you already have in your head.

A map is a machine that eats points

Start with the plane: an endless flat sheet, like a sheet of paper with no edges. Every point on it is pinned down by two numbers, a horizontal one and a vertical one.

A map is a rule that takes any point and sends it somewhere else on that same sheet. Feed it a point, get a point back. That is the whole idea.

One detail matters more than it looks. The rule takes two numbers in and hands two numbers back, the same count on both sides. That is what makes it reasonable to ask whether you can run it backwards at all. A rule that swallows two numbers and returns one has thrown something away before you have even started.

Drag the dot around and watch where its shadow goes. The grid underneath shows what the rule is doing to the entire plane at once, not just to your one point.

Notice that the grid bends but never tears. That is deliberate. We are going to be dealing with a very well-behaved family of rules called polynomial maps, where each output coordinate is built out of the input coordinates using nothing but addition and multiplication.

The rule in that figure is a fair example of one. It takes the point (x, y) and hands back (x + y²/4, y + x²/5). Square a number, scale it down, add it to the other one. That is the entire vocabulary. Fixed fractions like the quarter and the fifth are allowed, because dividing by a number is only multiplying by a different number. What is banned is dividing by a variable, or taking a square root, or anything that would take forever to compute. No 1/x, no √x, no sine waves. Just arithmetic, and only finitely much of it.

Polynomials are the tamest interesting objects in mathematics. They never shoot off to infinity anywhere you can point at, never have gaps or holes, never do anything sudden. Which is exactly why what is coming is strange.

They are also finite in a way almost nothing else is. A polynomial map is completely described by a short list of numbers, which means the entire universe of them is something you could, in principle, walk through one at a time. Hold on to that. It matters at the end.

The question is whether you can undo it

Some maps can be reversed. If I hand you the output, you can tell me the input, uniquely, every time.

Others cannot. If a rule sends two different points to the same place, then the output has lost information. Handed that output, you have no way to know which input it came from. The rule is a one-way street.

For polynomial maps, colliding is the only thing that can go wrong, and that is already a gift. Over the number system this conjecture actually lives on, which I will come to shortly, a polynomial map that never sends two points to the same place automatically covers the whole target plane as well, leaving no point of it unreached. And the rule that undoes it is automatically a polynomial too, rather than something uglier. You never have to ask for the undoing to be tidy. Tidiness comes free.⁴

So the whole question collapses down to a single word. Does the rule ever collide?

Given a polynomial map, how do you tell?

There is a beautiful answer, and it is a local one. Local is the word for anything you can settle by standing at one point and looking at a tiny patch of space around it. Global is the opposite: anything about the map taken as a whole. Hold on to that distinction, because the entire story turns on it.

Here is the local answer. Zoom far enough into any single point and a map built out of arithmetic stops looking curved. Bent grid lines straighten out. Whatever the map was doing in the large, up close it is only stretching space, squashing it, and tilting it. A rule that does nothing but those three things is called linear, and linear rules are about as easy as rules get.

You build that zoomed-in approximation by asking, for each output coordinate, how fast it responds to a nudge in each input coordinate. Two coordinates in and two out gives four such response rates, and they fit neatly into a small square table. That table is the Jacobian, named after the nineteenth-century mathematician Carl Jacobi.

Any square table of numbers has a standard recipe that crushes it down to a single number, called its determinant. Run the recipe on the Jacobian and you get the Jacobian determinant. The recipe itself does not matter here. What the number means does.

Think of that number as the map's stretch factor at a point. Draw a tiny square around your input. Look at the shape it becomes in the output. The Jacobian determinant is how much the area changed.

Move the square around. In most places it grows or shrinks a little. The stretch factor is a number that varies from place to place.

It can also go negative, which is not a problem. A negative stretch factor means the map turned the square over on the way through, the way a mirror does. The magnitude is the amount of stretching; the sign is only which way round it landed. Keep that in mind, because the number this whole story ends on is negative two.

Zero is where everything goes wrong

Now watch what happens when that number hits zero.

A stretch factor of zero means a tiny square got crushed into something with no area at all. The map flattened a two-dimensional patch down to a line. And when a patch gets flattened, points that used to be on opposite sides of it land on top of each other.

That is a fold. Like creasing a sheet of paper and pressing down: two layers, one position.

Drag the input point across the crease and watch the output turn around and retrace its steps. Two inputs, one output. The rule has become a one-way street, right there at the fold.

So folding is the enemy, and the stretch factor hitting zero is what folding looks like in arithmetic.

This gives you a very natural thing to demand. If you want a map that never folds anywhere, insist that its stretch factor is never zero.

One quiet shift before that demand can do any work. It is the only genuinely new idea I am going to ask you to take on, and the whole conjecture rests on it.

Everything you have dragged around so far lives on the ordinary plane, made of ordinary numbers, because that is what a screen can draw. The conjecture lives somewhere slightly roomier: the complex numbers. Those are what you get when you stop insisting that squaring a number must give something positive, and simply declare a new number, written i, whose square is negative one. Every complex number is then a pair, an ordinary part plus some amount of i, and all the arithmetic you know keeps working unchanged. Nothing breaks. You have only gained room.

The price is that you can no longer draw it. A single complex coordinate is already two ordinary numbers, so a map with two complex coordinates going in and two coming out is really a map of four-dimensional space. Every picture in this piece is a shadow of that, cast down onto the ordinary plane. The shadows are honest as intuition and I am going to keep using them. But the theorem needs the complex version, because of one property that ordinary numbers do not have.

Over the ordinary numbers, a polynomial can dodge zero forever. Take x² + 1. Squares are never negative, so it is never smaller than one, and it is obviously not a constant. Over the complex numbers that dodge disappears, because now there is a number whose square is negative one, and x² + 1 quietly factors into (x + i)(x − i). That is not a fluke about this one example. It is the fundamental theorem of algebra: every non-constant complex polynomial has a root somewhere.

And that forces the version we actually need, in any number of variables. Take a polynomial that is never zero anywhere, and follow it along any straight line you like. What you see along that line is a one-variable polynomial with no roots, so by the theorem above it has to be a constant. It never changes value along that line. Never changing along every line, in every direction, from every starting point, means never changing at all.

So over the complex numbers, a polynomial that is never zero has to be a constant. It has nowhere to hide. Which means "never folds anywhere" and "stretch factor is one fixed nonzero number, identical at every point in space" are not two conditions that happen to travel together. They are one condition wearing two different outfits.

That small miracle is what makes the question sharp enough to attack, and it is also, eventually, what makes the answer surprising. Maps that satisfy it have a name. They are called Keller maps, after Ott-Heinrich Keller, who wrote the question down in 1939.⁵

The conjecture

Here it is, in one line.

If a polynomial map never folds anywhere, can it always be undone?

That is the Jacobian Conjecture. For most of its life, the smart money was on yes.

You can see why. There is a theorem from calculus, the inverse function theorem, that says a nonzero stretch factor at a point guarantees you can undo the map inside some small patch around that point. Not the whole space. Just a patch. But a Keller map has a nonzero stretch factor everywhere, so it is undoable in a patch around every single point, without exception.

The conjecture asks whether all those local undoings glue together into one global undoing. It sounds like bookkeeping. It sounds like the kind of thing that should follow from a careful argument about how the patches overlap.

It does not follow. And here is the thing that should have warned everyone.

That map has a nonzero stretch factor at every single point, so it is undoable in a patch around every one of them. Taken as a whole, it is a disaster.

What it does is turn vertical motion into rotation. Walk the input straight up and the output travels around a circle, so after one full turn the output is exactly back where it began while the input is somewhere it has never been. Keep walking and a third input lands on that same output, and a fourth, forever. The map wraps the plane around itself infinitely many times. Local and global are just genuinely different questions.

The reason nobody panicked is that this example is not remotely algebraic. It is built out of an exponential, which is not a polynomial and does not obey any of the rules that make polynomials rigid. It grows without limit in one direction and shrinks toward zero without ever arriving in the other, so it has infinite room to work in. It misses the origin entirely, so its image is the plane with a puncture in it, and over every point it does hit it stacks infinitely many inputs. It has no degree at all, because writing it out takes an endless sum.

A polynomial can do none of that. Give a polynomial map a nonzero stretch factor and, above a typical output, only finitely many inputs land there. Its image fills everything except a thin sliver of the target. Its degree puts a hard ceiling on how many times it can wrap. Rigidity was the whole hope: that being a polynomial was stiff enough to close a gap a function like the exponential can obviously stroll through.

For one variable, that hope is correct, and the miracle from a moment ago proves it in a single line. With one coordinate in and one out, the Jacobian table has just one entry, so the stretch factor is that single response rate and nothing else. Never folding means that rate is never zero. A complex polynomial that is never zero has to be a constant. So the rate is a constant, which makes the map a straight line, and straight lines reverse.

For two variables and up, it was open for 87 years.

Eighty-seven years of nearly

The history is not a story of neglect. It is a story of a problem that kept looking solvable.

People proved enormous amounts about it. The pattern is always the same. Pick some way of saying "this map is not too wild," and for maps of that kind the conjecture stops being a conjecture and becomes a theorem. If the map has degree two, it is true. If the map is birational, meaning one-to-one everywhere except on a thin exceptional set, true. If it is Galois, meaning its collisions are perfectly symmetric, true. If it is proper, meaning nothing is able to slip away to infinity, true. Hold on to that last one. It is the door the counterexample eventually walked through.

In 1982, Bass, Connell and Wright proved a genuinely shocking reduction: if you allow yourself to add extra variables, the entire conjecture in every dimension collapses to the single case of cubic maps. Degree can be traded away for dimension. Whatever baroque high-degree monster you imagine might break the conjecture, there is a tame-looking cubic one, in more variables, that breaks it just as well.⁶

The conjecture also turned out to be entangled with famous problems that look nothing like it. It is equivalent, in a precise sense, to the Dixmier Conjecture, which asks whether every self-map of the Weyl algebra of quantum mechanics can be undone, and to the Poisson Conjecture, its counterpart for classical mechanics. Nobody had any right to expect that a question about undoing arithmetic would turn out to be the same question as one about position and momentum. It is.⁷

And it broke people. It is famous for the number of published proofs that turned out to contain subtle errors. In November 2004 a proof of the two-variable case was announced with a prominent algebraist initially vouching for it. It was retracted within weeks. The scheduled follow-up lectures never happened.⁸

There was also a near miss that reads, in hindsight, like a rehearsal. In 1994 the real-number version of the question fell, in two variables, to a map that misbehaves in almost exactly the way Alpöge's does. It did not touch the complex conjecture, for a reason that gets its own section later. But from 1994 onward, anyone paying attention knew what a counterexample would look like if one existed.

Two things are worth saying honestly here. First, the specialists were not unanimous. For dimension three and up, several of the people closest to the problem had gone on record saying they did not believe it. Arno van den Essen wrote in 1997 that there was an enormous difference between two variables and three. T. T. Moh wrote that there was hardly any evidence for the higher-dimensional versions being true at all.⁹

Second, and this is the part that matters: nobody found one anyway. For 87 years, across a field full of people who suspected the thing was false, not one person produced a counterexample.

The counterexample

Three coordinates in, three coordinates out. The stretch factor is exactly negative two, at every point in space, forever. It never folds. It cannot fold.

And it sends three completely different points to the same place.

So the answer to the question is no. A polynomial map can refuse to fold anywhere and still be impossible to undo. That is the whole disproof. It fits in a tweet, and there is nothing exotic inside it: no strange constants, no fractions, every coefficient a whole number and not one of them bigger than twelve. You can check it in half a minute in any computer algebra system, and a great many people did within hours. There is no limiting argument, no approximation, no delicate estimate a referee has to take on trust. Take the derivatives, take the determinant, get negative two. Plug in the three points, get the same answer three times. Done.¹⁰

The consequence propagates upward immediately. Pad the map with untouched extra coordinates and you have a counterexample in four dimensions, five, any number. And because a 1983 result of Connell and van den Dries says the conjecture holds for every field of characteristic zero or none of them, this single map over the rationals kills it everywhere.¹¹

The plane case, two variables, is still open. That is not a footnote. It is the case Keller's predecessor originally asked about in 1884, the case with the most evidence behind it, and it survived. It is known to hold for every map of degree up to a hundred, which means any counterexample still hiding down there is a vastly larger object than the degree-seven map that just killed everything above it.¹²

How does it not fold?

This is the part that should still bother you. If three inputs land on one output, they have to merge somehow. And merging is folding. And folding is forbidden.

The resolution is beautiful, and it is the real content of the counterexample.

First, a word for the thing we keep talking about. Fix an output point and ask which inputs land on it. Those inputs are called its preimages. A fold is what happens when two preimages drift together, touch, and become one, and that is the only way most people ever picture the count going down.

The three preimages of Alpöge's map never merge. Not anywhere, not over any target. When you move the target to a place where fewer than three of them exist, the missing ones do not slide together and cancel out. They run away to infinity.

You can watch exactly that in the counterexample figure above. Drag its target slider toward zero and two of the three preimages slide off the ends of the track and are simply gone, while the closest pair of them only ever gets further apart. Three becomes one, and nothing ever touches.

Here is the same trick in one dimension, where every moving part is visible at once.

A cubic has three roots. Flatten it, by shrinking the coefficient on its cube term toward zero, and it degenerates toward a parabola, which has two. One crossing has to go.

Watch how it goes. It does not drift into a neighbor and annihilate with it. Its neighbors are not even close, and the gap between the two of them never narrows. It slides out along the curve, past the edge of the frame, and when the coefficient passes through zero it reappears on the far side.

That is the loophole, stated precisely. A nonzero stretch factor controls every patch of space you can draw a box around. It says nothing whatsoever about what happens to something that keeps going and never stays inside any box at all. The map is not proper. A preimage is allowed to leave through infinity, so the count of them can change without anything ever folding.

Everything in the conjecture was about local behavior. The counterexample lives entirely at infinity, where local behavior has no jurisdiction.

And the loophole was not a secret. Over the ordinary numbers it had already been used, thirty-two years earlier, in the very dimension where the complex question is still open today. In 1994 Sergey Pinchuk wrote down a polynomial map of the real plane whose stretch factor is strictly positive at every single point, and which sends two different points to the same place anyway. It works by letting a preimage escape, exactly the way Alpöge's does.¹³

Nobody called that the end of the Jacobian Conjecture, and they were right not to. Pinchuk's stretch factor is positive everywhere without ever being constant, which over the ordinary numbers is a perfectly legal thing to be. This is x² + 1 again, the polynomial from earlier that dodges zero forever without settling down. Over the complex numbers that dodge closes instantly, and the stretch factor of a never-folding map is forced to be a single frozen constant. The miracle that made Keller's question sharp is the same miracle that made it hard, and it is the reason a real counterexample in 1994 left the complex one standing for another thirty-two years.

So the crack had been in plain view for a generation. Everyone knew the shape of the room a counterexample would have to live in. What nobody could do was build one rigid enough to survive the crossing into the complex numbers.

Why three?

The choice of three is not arbitrary either, and this is where you can see the shape of the idea rather than the algebra.

Count upward and watch the options vanish.

If the map is one-to-one, that is the conjecture holding, so that is out by definition. A counterexample has to send at least two inputs to some output. Picture those inputs as sheets stacked above the output, one sheet for each preimage, and ask how many sheets the smallest possible counterexample needs.

Two is out, for a subtle and rather pretty reason. With only two sheets there is exactly one way to shuffle them: swap. A two-sheeted map therefore has a perfect symmetry whether it wants one or not, which is precisely what being Galois means, and the Galois case had been settled long before. Keller himself had already ruled out the birational maps, the ones that are one-to-one everywhere except on a thin exceptional set.

Three sheets is the first arrangement that can be genuinely lopsided. There are shuffles of three things that leave one of them alone, so a three-sheeted map is under no obligation to be symmetric. It was the smallest thing left standing.

And there is a very natural three-to-one map sitting in plain sight.

Take a cubic with three roots. Pick one root to be special and group the other two together. There are three ways to do that, and they are genuinely different, but they all rebuild the same cubic when you multiply them back together. So the map from "one root plus a pair" to "the cubic" is three-to-one, for free, by pure symmetry.

The counterexample is what you get when you cut that construction down to size. As described, it does not yet live in three-dimensional space, so you slice it with exactly the right hyperplane, and the slice you land on turns out to be an ordinary three-dimensional space in disguise, which is what lets the whole thing be written as three polynomials in three variables. Terence Tao, who wrote up the geometric explanation, says plainly that he cannot fully explain why that last step works.¹⁴

"It is just a counterexample"

The result landed in the middle of a live argument about what machines are doing to mathematics, and it got read through that argument immediately.

The criticism has a few shapes. That a counterexample is not a theory, and does not build the machinery that makes a proof valuable. That the higher-dimensional conjecture was widely doubted anyway, so this confirmed a suspicion rather than overturning a belief. That a language model is a very good pattern matcher and this was, underneath, a search. That a human expert pointed it at the problem, so the interesting judgment was still human.

The first two are fair and I would not argue with them. Tim Gowers, who called the result amazing, was careful to note it was a counterexample and therefore not the end of anything.¹⁵ Daniel Litt said the same and added, cheerfully, that he does not much care about this particular conjecture.

The search argument is the one that does not survive contact with the numbers.

Here is the shape of the problem the machine was actually facing. Writing down a polynomial map means choosing its coefficients, and those coefficients are the knobs you get to turn. Demanding that the map never fold does not cost you one condition; it costs a whole pile at once, because the stretch factor is itself a polynomial, and forcing a polynomial to be constant means forcing every one of its non-constant coefficients to zero, separately. Knobs on one side, conditions on the other. The figure counts both.

The map has degree seven. Its Jacobian determinant, before you impose anything on it, is a polynomial of degree up to eighteen, and a polynomial of degree eighteen in three variables has 1,329 non-constant coefficients. Every one of them has to vanish. Against that, a general degree-seven map in three variables gives you 360 coefficients to choose.¹⁶

You are being asked to satisfy 1,329 conditions by turning 360 knobs. Systems shaped like that have no solutions at all, almost always. There are not enough knobs, and the conditions start contradicting one another long before you run out of things to try.

The figure has a second panel underneath it, and it is worth a minute of your time. It drops all the way down to the easiest interesting case, two variables at degree two, where there are only six coefficients and five conditions, and it hands you the six sliders. Try to switch all five conditions off at once. Killing the first two is easy. Holding those two dead while the other three also go to zero is the entire difficulty of this subject in miniature. If you manage it, you have built a Keller map by hand, and you can then go looking for a point it hits twice. You will not find one, because at degree two the conjecture is a theorem. Now imagine the same exercise with 1,329 rows instead of five.

The map exists anyway, and that is the part worth sitting with. Every undoable polynomial map is automatically a Keller map, for a one-line reason: if two maps undo each other, their stretch factors have to multiply to one at every point, so neither is ever allowed to be zero. So Keller maps are not rare in the sense of barely existing. You can manufacture them all afternoon by composing simple pieces. They are rare in the sense of being nowhere near where you would look: a thin, highly structured sliver threaded through a 360-dimensional space. And every point of that sliver anyone had ever written down turned out to be undoable. Landing on the sliver is not the achievement. Landing on a spot where the map does not come back is.

Tao ran the same arithmetic and concluded that a polynomial like this looks highly unlikely to be reachable by brute force. Bartosz Naskręcki put it more directly: it requires real insight. Abhishek Saha, who verified it, said that searching everything would not have worked, so obviously there was insight involved, and that insight has not been published.¹⁷

That is the honest state of it. We know a machine found it. We do not know how, because the reasoning trace was never released. That is a real gap, and it is the most legitimate complaint in the whole discourse.

What actually changed

Set the machine question aside for a second and look at what changed in mathematics.

Three named conjectures fell at once. The Jacobian Conjecture for every dimension above two, and with it the Dixmier and Poisson Conjectures, which had been proven equivalent to it. All three now survive only in the low dimensions the counterexample does not reach. Decades of partial results did not become worthless, but they did change meaning: every theorem of the form "the conjecture holds for maps of this type" is now a list of properties that a counterexample is forbidden to have.

And it turned generative almost immediately. Within four days of a 2 a.m. tweet, papers were on arXiv. A cubic-homogeneous counterexample in 79 variables, extracted by pushing the map through the Bass-Connell-Wright reduction. A separate counterexample to a repaired characteristic-two version of the statement. A paper determining exactly which maps in the deformation family around this one, over the real numbers, fail to have dense image. A result showing that Keller maps with a certain grading can never be counterexamples, which is a new obstruction discovered by staring at why this one works.¹⁸

That is what a good counterexample does. It does not just close a question. It hands the field a concrete object to take apart, and the taking apart is where the theory comes from.

Kevin Buzzard, who has spent years arguing that formal verification and machines would matter to mathematics before most of his colleagues agreed, wrote up his reaction the same day. He mentions a colleague dismissing an earlier machine-found counterexample on the grounds that its easiness just showed humans had not spent enough time on the problem. Buzzard's response was that his colleague seemed to be working through the five stages of grief, and was currently in denial.¹⁹

I think the dismissal and the hype are making the same mistake. Both are reading this as a referendum on machines.

Here is what actually happened. A question that a hundred serious people could not answer in 87 years was answered in an afternoon, and the answer was small enough to read in one breath. Remember that a polynomial map is only a short list of numbers. The right list had been sitting in the space of degree-seven maps the entire time, no more hidden than any other point in it. Keller could have written it down. Any of the people who published flawed proofs could have written it down. Nobody did.

Something changed about what is reachable. Not about what is true.

The conjecture was never really about polynomials. It was about whether controlling something everywhere locally controls it globally, and the answer is no, and the reason is that infinity is a place where things can leave. That was always the answer. It just needed someone, or something, to look in the right direction for one afternoon during a World Cup final.

The plane case is still open. Somebody should probably ask.