The Jacobian Conjecture for N = 2

There is a folder in one of my repositories that I keep coming back to. It holds a research workspace for a single open problem: the two-dimensional complex Jacobian conjecture, which everyone in the area writes as JC(2). Thirty files, a dozen exact verifier scripts, and two ledgers whose only job is to stop me from believing my own conclusions.

I want to walk through what is in it. Not because I have solved anything — I have not, and neither has the corpus — but because the shape of a long, unfinished problem is worth seeing. Most of what follows is other people’s mathematics, carefully sourced. The parts that are ours are marked as such, and the parts that failed are kept in the open where they can keep teaching.

The statement that refuses to close

Take a field $k$ of characteristic zero and a polynomial map

The Keller condition is that its Jacobian determinant is a nonzero constant:

That is a strong demand. It says the derivative of $F$ is invertible at every point, with no degeneration anywhere, not even far away. The conjecture asserted that this forces $F$ to be a polynomial automorphism — invertible, with a polynomial inverse.

In one variable it is a one-line argument: $f’(x)=c\ne 0$ gives $f(x)=cx+b$. In three or more variables it is false. What remains is exactly one case:

There is no loss in normalizing the constant: if $J(P,Q)=c$, replace $Q$ by $Q/c$ and assume $J(P,Q)=1$. In differential-form language the same condition reads $dP\wedge dQ=dx\wedge dy$, and in Poisson-bracket language it is simply

Two variables. One equation. Open since 1939.

The three-variable counterexample, and why it does not rescue us

The reason JC(2) is the last fixed-dimensional case is that somebody found a map in three variables that satisfies the Keller condition and is still not injective. The corpus keeps it written out, because a checkable object beats a citation:

Exact symbolic differentiation gives $\det JF\equiv-2$, everywhere, as an identity in the polynomial ring. And the map collides:

Three distinct source points, one image. Those are two logically independent certificates — the first is a global identity, the second a finite collision — and appending identity coordinates pushes the failure into every dimension above three.

So why does this not settle the plane? Because of how it works. On the chart $x\ne0$, substitute the root coordinates

and the map becomes something transparent: $t$ is a root of the cubic

while $r=3ct^2-4t+B$ recovers the remaining coordinate. The construction remembers one marked root of a cubic and routes the derivative factor that would otherwise obstruct it through an independent scale direction. That third direction is not decoration. It is the resource.

In dimension three, a derivative factor can be routed through an independent scale direction while two shape variables keep their multi-sheet geometry. In dimension two, every attempted routing has to be reconciled on a boundary lattice or on a high-genus punctured fiber.

A proof of JC(2) must show that reconciliation is impossible in every degree. A counterexample must exhibit the first boundary architecture where it succeeds. Everything below is one of those two errands.

What would actually count as an answer

It is worth being precise about the finish line, because a lot of near-misses fail here rather than in the mathematics.

A counterexample is finitely certifiable. The ideal artifact is explicit $P,Q\in\mathbb Q[x,y]$ with $J(P,Q)\equiv1$, plus explicit distinct points $p\ne q$ with $(P,Q)(p)=(P,Q)(q)$ — or a rigorous field-degree or non-surjectivity certificate in place of the collision.

A proof is harder to certify honestly. It has to be uniform in degree and cover arbitrary planar Keller maps. Eliminating weighted, symmetric, cubic-root, or degree-bounded families is not enough unless a universality theorem forces every hypothetical counterexample into the eliminated class. That missing universality step is where most of the interesting partial results in this folder actually sit, and saying so out loud is most of what keeps the corpus useful.

Counting sheets: the field-degree ladder

The single most organizing invariant is the generic field degree. Write $K=\mathbb C(P,Q)$ inside $L=\mathbb C(x,y)$ and set

This is the number of points in a generic fiber, counted before ramification — how many sheets the map has. $N=1$ is the birational case, and there the Keller condition forces invertibility. So a counterexample needs $N\ge2$, and the last forty years have been a slow climb up that ladder:

• $N=2$ falls to the finite-normalization and Galois argument.
• $N=3$ is excluded by Orevkov’s three-sheet theorem.
• $N=4$ is excluded by the Domrina–Orevkov boundary-diagram theorem.
• $N=5$ is excluded by Żołądek.

So any counterexample satisfies

That last rung is where this corpus learned a lesson about itself. For a long stretch the workspace was organized around degree $5$ as the first open case, with two entire files and a long chain of numbered findings devoted to the dihedral $D_5$ frontier. An independent audit — run by an AI model, over the whole corpus, with instructions to check citations rather than extend them — found the Żołądek result in a published reference list and reported the consequence plainly: the frontier was $N\ge6$, and those two files were beautiful mathematics about an empty case.

Nothing in them was wrong. They were simply aimed at a case that no longer existed. That correction was worth more than the new lemmas that came with it, and it is the reason the ledger discipline described at the end of this post exists.

The normalization picture

Here is the structure almost every serious attack uses. Put

and let $S$ be the integral closure of $A$ inside $L$ — the finite normalization. Geometrically, $X=\operatorname{Spec} S$ is a normal affine surface carrying a finite map $\pi:X\to\mathbb A^2$ of degree $N$, and the original plane sits inside it as an open subset whose complement is a boundary divisor $D$. In fact JC(2) is equivalent to the statement that no such $X$ exists with $N\ge2$ and an open embedding $\mathbb A^2\hookrightarrow X$ making $\pi$ étale on that open part.

This is a good trade. A Keller map is not a finite morphism — that is the whole problem — but $S$ is finite and flat over $A$, free of rank $N$ by Quillen–Suslin, and normality lets codimension-one valuations control everything. All the failure gets pushed into the boundary $D$, where it becomes divisor combinatorics.

Two structural facts survive the trade. Since the map is étale on the affine part, every branch curve downstairs must be met by both an unramified affine sheet and a ramified boundary sheet — the mixed-sheet condition, which is the real bridge from the Keller hypothesis into finite group theory. And the normalization of a counterexample cannot be globally monogenic, cannot have a principal different, and cannot be Galois: each of those would force invertibility outright.

The Galois closure of $L/K$ carries a monodromy group $G$ acting transitively on $N$ sheets, generated normally by the inertia of the boundary divisors. At $N=6$, exactly $6$ of the $16$ transitive subgroups of $S_6$ survive the inertia filter — an exact computation, not a sample.

A budget at the boundary

The sharpest external input is a corollary of Orevkov’s that the corpus had cited for years without using. If $e_i$ is the ramification index along boundary divisor $D_i$, then

That single inequality cascades. With $k$ the number of boundary components and $b$ the number of ramified ones:

Every ramified boundary divisor costs at least one unit of a budget that never grows with degree — and the class group of the normalization is free on the boundary components, so the same bound caps its rank.

Then comes the honest limitation, which was found by trying to break the result rather than to extend it. The obvious hope is that high degree needs more ramification than the budget allows. It does not. A transposition costs exactly $2$ units for every $N$, so a hypothetical counterexample can keep its ramification bounded while its degree runs away. The budget is real and it is not asymptotically binding. Recording that as a negative result rather than leaving it as an open direction is, I think, the single highest-value habit in the whole workspace.

A residue-weighted refinement sharpens the same idea, with $f_i$ the residue degree and $\kappa\ge0$ an Orevkov defect:

and a collision bound says that if a point of an image curve has $q$ normalization preimages then

In degree six with an irreducible nonproper set, those two together cut the monodromy group down to $A_6$ or $S_6$. That is a very small target, and it still has not been closed.

Where the map fails to be proper

A noninvertible Keller map cannot be proper, and its failure is visible in the target as the nonproper-value set $A_F$: the points approached by images of source points escaping to infinity. A counterexample exists exactly when this set is nonempty, which makes it the most Keller-specific object available.

The corpus’s own contribution here is a clean identification, $A_F=\pi(D)$ — the nonproper set is precisely the image of the boundary. Combined with the budget above:

Externally, $A_F$ is known to have all of its branches meeting the line at infinity at one point, and a polynomial Vitushkin argument shows that a counterexample’s nonproper set cannot be homeomorphic to $\mathbb C$. So the curve must have a self-intersection or several branches — for an irreducible $A_F$, its parameterization is provably non-injective, a genuine self-crossing rather than a cusp.

On a generic fiber the bookkeeping becomes an identity. With $\Lambda$ the total ramification, $g$ the genus of the fiber’s smooth completion and $r$ its number of punctures,

proved three independent ways in the corpus, and later sharpened: separating finite-valued punctures from poles on an adapted fiber gives

which forces $r\ge3$ unconditionally.

One component at a time

Everything so far constrains the pair $(P,Q)$ or the surface $X$. The most recent turn constrains one component on its own, and it is my favorite thing in the folder.

Fix a Keller component $P$ and consider the Gelfand–Leray form on a fiber $Y$ of $P$:

The Keller condition says exactly that $\omega$ is regular and nowhere zero on every component of every fiber — and that the partner $Q$ restricts to a regular primitive of it, $dQ|_Y=\omega|_Y$. In other words, $Q$ is the flow time of a form determined by $P$ alone.

That has real teeth. If $P$ is a Keller component then $\omega$ must be exact on every fiber component: all residues and all periods vanish. And the degree itself turns out to be computable from $P$ without ever mentioning $Q$:

so total pole order $1$ means automorphism, and a counterexample needs pole order at least $6$. Applied to a family that looks perfectly innocent — nowhere-vanishing gradient, smooth fibers, irreducible generic fiber — it kills all of it at once:

The pleasant irony is that the corpus’s own long-standing “hard counterexample”, $x(1+xy)$, turns out not to be a Keller component either. It had been blocking a line of attack for months by pure misclassification.

Degree six, braids, and models that saturate everything

The current frontier is where I find the honest character of the problem clearest. In the last remaining sextic branches, the surviving arithmetic is so tight that people can write down curves matching every numerical budget at once — plane degree, genus, conductor, ramification $5$, defect $3$, pole band, collision type — and then watch them die for a reason no counter could see.

A degree-ten rational curve realizes the complete local, plane, and pole ledger; its auxiliary degree-five projection then fixes no transitive marking, and it is gone. An exact rational sextic saturates both the genus and the ramification–defect ledger; its infinity braid fixes none of the complete transitive Nielsen classes of sizes $15625$, $20480$, and $21870$, and it is gone. Exhausting all $17$ genus-allowed one-place sextic types leaves only smooth infinity, and a complete signed census of self-carousels closes that too.

The lesson repeats with unusual regularity: the scalar budgets are sharp, and sharp is not the same as sufficient. Every time the numbers alone are made to fit, the obstruction moves one level up — into full braid factorization, or into whether the configuration is realizable by an actual plane curve at all.

The coefficient campaign

A parallel line attacks coefficients directly. The first vertical profile not removed by the audited degree screen is

Writing the leading coefficients as $a_6=\alpha h^2$ and $b_9=\beta h^3$, the first negative fractional-root coefficient forces an exactness condition on a cube root of $h$:

Squarefree nonconstant $h$ dies immediately. Repeated $h$ collapses into a classical Moh/Formanek–Stothers polynomial differential equation plus a finite Belyi passport problem, and the full bracket system over $\mathbb C(x)$ reduces to

Polynomiality and $3\mid\deg H$ force $H$ to be a cube, which removes the entire noncube branch — including all nine enumerated low-degree passports — without assuming polynomial descent of the approximate roots. Inside the surviving cube branch, rationality forces a nonconstant leading cube root to be a single-root power $g=\lambda(x-b)^s$ with $s\ge2$, and a long sequence of tropical and weighted-boundary arguments closes fiber after fiber of the coefficient tail.

These are real exclusions, verified by exact enumerations running into tens of thousands of cells. They are also bounded: no theorem yet places every hypothetical Keller map into the vertical $(6,9)$ profile. Which is the same missing universality step as before, wearing different clothes.

The degree wall, and the certificate that is not here

The published Newton-polygon program pushes any planar counterexample to maximum degree at least $108$, and below maximum degree $125$ leaves exactly one surviving degree pair and its transpose:

An external note reports exact ideal-membership certificates excluding the two residual systems, which would raise the bound to $125$. The corpus records the equations, the construction dimensions, the file hashes, and the reported output — and states plainly that the roughly 89 MB certificate archive is not present locally and cannot be replayed.

I like that entry more than most of the proofs. It is a result recorded at exactly the confidence its evidence supports, with the boundary of that evidence written down next to it.

The part I actually reuse

Strip away the algebraic geometry and there is a working method in that folder that I have since used elsewhere, in projects with nothing to do with polynomials.

Two ledgers, not one. A claim ledger says which mathematical statements may be used as premises, with statuses — established, proved here, computationally verified, plausible, open blocker, disproved, dormant, superseded. An idea registry separately says which research mechanisms are active, dormant, superseded, or withdrawn. They answer different questions. Conflating “this statement is false” with “this direction is not worth funding” is how programs quietly lose ideas that were only mis-executed. Anything unmapped into either ledger is unclassified, and unclassified never gets treated as a theorem.

A circularity filter. Some appealing targets have the form: if condition $\mathcal C$ holds, the map is invertible. Before spending months on one, check the converse. If invertible also implies $\mathcal C$, then the theorem is a kill switch, not a reduction — its hypothesis is just the conjecture again, in costume. Kill switches are perfectly good counterexample filters and useless as proof targets. Applying that one test reclassified several of the corpus’s own “active closure targets” in an afternoon.

Negative results are results. The budget’s failure to bind asymptotically, the numerical system that turned out feasible with $5304$ integer solutions, the Euler-characteristic obstruction that reduced to $0=0$ — each is written down as a finding with its own identifier. They are the cheapest thing in the folder and they save the most time.

Retractions stay visible. When a lemma named the wrong pencil and inferred a false degree, the fix was not to quietly edit it. The wrong version stays, marked, with the correction beside it and a note on which downstream results survive. Anyone who reads it learns the failure mode too.

Every finite claim gets an executable check. Eleven verifier scripts, each printing a single pass line, covering identities, budgets, monodromy over all transitive subgroups of $S_4$, $S_5$, $S_6$, fibration and projection facts, and the Gelfand–Leray arithmetic. A claim with a finite symbolic component and no verifier does not get promoted.

Where it stands

JC(2) is not proved and not disproved. A counterexample needs $N\ge6$ sheets, non-Galois monodromy, a nonprincipal different at the deleted boundary, mixed affine and boundary sheets over every branch curve, at most $N-2$ boundary components, a nonproper curve that is not homeomorphic to $\mathbb C$ with all branches through one point at infinity, and a generic fiber of genus at least $2$ under the standard reduction. That is a remarkably specific ghost. Nobody has found it, and nobody has proved it cannot exist.

What strikes me, after enough time with this material, is how much of the work is not deduction at all. It is bookkeeping under uncertainty: keeping the established separate from the plausible, the empty case separate from the open one, the sharp bound separate from the sufficient one. The mathematics is out of my depth on most pages. The discipline is not, and it turned out to be the transferable part.

The next real move is unglamorous and clearly stated: get the two remaining braid frontiers to full nonabelian factorization, extend the coefficient exclusions past their current fibers, and — above all — find the coordinate-invariant bridge that carries an arbitrary Keller map into one of the controlled models. Without that bridge, every exclusion in the folder stays exactly what it is: true, exact, and bounded.