Charting the unknown

On the evening of 19 July 2026, Levent Alpöge posted a polynomial map of three variables and credited its discovery to Claude Fable 5:

$$F=\big(AB,\;\;y+3xB,\;\;2x-3x^2y-x^3z\big),$$ $$A=1+xy,\qquad B=A^2z+y^2(4+3xy).$$

Its Jacobian determinant is $-2$ at every point. So at every point the map is locally invertible, with no crease or fold anywhere. Yet the three points $(0,0,-\tfrac14)$, $(1,-\tfrac32,\tfrac{13}2)$ and $(-1,\tfrac32,\tfrac{13}2)$ all land on $(-\tfrac14,0,0)$. Keller had asked in 1939 whether a polynomial map with constant nonzero Jacobian must have a polynomial inverse. Eighty-seven years later, the answer in dimension three is no.

This essay walks through what that answer did to the mathematics around it. It is also a demonstration of an atlas: every claim below is a page you can open, every relation between pages is a sentence someone has to argue, and the pictures are computed from those sentences. The network beside the text moves as you read.

A map you can check

The first thing an atlas owes its reader is a way not to take its word for things. This box is not a picture of a verification. It is one, run by the server when this page was sent, in exact rational arithmetic, and it can be run again in your browser:

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Alpöge’s map of ℂ³ has constant Jacobian determinant −2 and sends three points to one, so it is not injective: the Jacobian conjecture fails in dimension 3.

• ✓ the map has total degree 7 degrees 7, 6, 4
• ✓ det JF is the constant −2, identically det JF = -2, expanded exactly over ℚ
• ✓ three distinct points (0, 0, -1/4), (1, -3/2, 13/2), (-1, 3/2, 13/2)
• ✓ all three have the same image F = (-1/4, 0, 0) at each
• ✓ a second fibre: three more points with one image (1, -2, 9), (-1/3, 4, 27), (-2/3, -1/2, -9/8) ↦ (-1, 1, -1) (Fong–Fable ledger C4)

The fifth line is a second collision, found independently on 20 July in the Fong–Fable session at jacobianconjectures.com. The map has also been checked in Isabelle/HOL and in Lean, against a statement pinned to a commit, so it carries the atlas's strongest tier, V0. Most of what follows is weaker than that, and the atlas says so every time. A refereed theorem is V3. A preprint, a post or an unrefereed proof is V4, a claim, and the figures draw claims as claims.

Why three

A general point of ℂ³ has exactly three preimages under $F$. Three is the least possible number. Keller's own theorem rules out one: a Keller map with one preimage over a general point is an automorphism. Campbell's theorem rules out two: a two-sheeted cover is Galois, and a Galois Keller map is an automorphism. The Fong–Fable ledger recorded this exclusion on 20 July, and the same day Alexis Gallagher published a family with every degree from three up. The degrees that étale self-maps of ℂ³ can have were settled within a day of the question being asked: 1, and every integer from 3.

The chart for ℂ³

The chart is worth remembering. Later there will be the same chart for the plane, and it will be empty.

How it works

Why should such a map exist? D. Speyer's account is the clearest. Take a curve in the plane and sweep its tangent lines. The map $(w,\gamma)\mapsto(p(w)+\gamma p'(w),\ q(w)+\gamma q'(w))$ sends a point of the curve and a distance along its tangent to the point you reach. After a reparametrization the Jacobian is simply $\gamma$. A general point of the plane lies on as many tangent lines as the curve's dual has degree, so it has that many preimages. Alpöge's three preimages are three tangent lines through one point. S. Gao turned the idea into counterexamples in every dimension from three, with the degree as large as you like.

The map also has a symmetry. Scale $x$ by $t$, $y$ by $t^{-1}$ and $z$ by $t^{-2}$, and the three components scale by $t^{-2}$, $t^{-1}$ and $t$. T. Shaska proved that such a symmetry must have weights of both signs, and that it needs at least three variables. In the plane, any nontrivial torus symmetry forces a Keller map to be an automorphism. The symmetry is where the construction lives, and the plane has no room for it.

Dividing out by the symmetry leaves a problem in two variables, $t=xy$ and $w=x^2z$. There the whole Keller condition becomes one identity in the Poisson bracket. The Fong–Fable ledger wrote it down on 20 July, and Shaska derived it independently. The atlas checks it too:

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Alpöge’s map descends to the invariants t = xy, w = x²z, and there its constant Jacobian becomes one bracket identity with constant 2 (the master equation of the Fong–Fable ledger).

• ✓ F₂F₃ = m·g̃ with t = xy, w = x²z polynomial identity in x, y, z, exact
• ✓ F₁F₃² = m²·h̃ polynomial identity in x, y, z, exact
• ✓ m{g̃,h̃} + 2h̃{g̃,m} + g̃{m,h̃} is the constant 2 the bracket expression = 2

The ladder

The Jacobian conjecture was never alone. It has relatives indexed by dimension, joined to it by theorems that shift the index. The Dixmier conjecture asks the same question of the Weyl algebra of differential operators. It implies the Jacobian conjecture in the same dimension, and is implied by it in twice the dimension. The Poisson conjecture sits between them. The Hessian conjecture is the Jacobian conjecture for gradient maps, and it implies the Jacobian conjecture in half its dimension. Draw each family as a row and each dimension as a column, and the theorems become arrows:

Only a few of these cells are declared on the family pages, and every declaration there cites its evidence. The atlas derives the rest from the arrows and records each derivation's chain, which the cell's tooltip shows. Red enters at JC₃, with Alpöge's witness, and spreads right because padding by the identity carries a counterexample up a dimension. It climbs to DC₃ along the easy implication, and the explicit witness there is the pullback of Alpöge's map to differential operators. It reaches HC₆ by doubling. HC₅ is Meng and Yang's own counterexample, a polynomial of degree 14 in five variables. Their paper is unrefereed, so the atlas checked it itself:

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Meng–Yang’s polynomial Ψ in five variables has constant Hessian determinant 128, yet its gradient map sends two points to one: the Hessian conjecture fails in dimension 5.

• ✓ Ψ has degree 14 42 monomials, integer coefficients
• ✓ det Hess Ψ is the constant 128, identically det Hess Ψ = 128, expanded exactly over ℚ
• ✓ two distinct points (1, -3/2, 0, 0, 0), (-1, 3/2, 0, 0, 0)
• ✓ both have the same gradient ∇Ψ = (0, 0, -1/2, 0, 0) at each

What stays hollow is a single chain. DC₂ implies JC₂, JC₂ implies DC₁, HC₄ implies JC₂, and PC₁ is JC₂ restated. Dixmier's original question from 1968, the first Weyl algebra, now stands or falls with the plane. The dashed cells are claims. Zheglov's proof of DC₁, Long's counterexample to PC₂ and Wilson's proof of the two-variable Gaussian moments conjecture are written but not yet adjudicated, and the ladder shows them as claims, not as results.

Dominoes

Several conjectures had been proved to imply the Jacobian conjecture: Zhao's vanishing conjecture, the image conjecture, Mathieu's conjecture about compact groups, and the Gaussian moments conjecture. When the Jacobian conjecture fell, each of them fell by implication. That kind of falsity is thin. It says a counterexample exists, in some dimension, and does not hand you one. The weeks after July were largely spent turning implications into objects.

W. Thompson ran the classical Bass–Connell–Wright reduction on Alpöge's map by hand and got a cubic-homogeneous counterexample in 24 variables. It was cut to 20 and then 19. The symmetric construction of de Bondt and van den Essen doubled those into gradient counterexamples in 48, 40 and 38 variables. Prellberg states that the 40- and 38-variable ones explicitly break Zhao's vanishing conjecture. None of these is refereed, and the atlas has not re-run them.

Some witnesses came from elsewhere. Christopher Long found counterexamples to the Gaussian moments conjecture in three variables, using no Keller map at all, and reports a Lean proof. A. Dvorsky gives an explicit counterexample to the generalized vanishing conjecture in five variables, and Long gives one to Mathieu's conjecture on SU(2). Both are preprints. Whether Mathieu's conjecture fails for SU(N) in every dimension from three, or only from 24, turns on how one reads a 1997 implication. The atlas records both readings and names who holds each.

What is left

Here is the same chart as before, for the plane. Every band is a theorem saying no counterexample lives there:

What is left of JC₂

Any counterexample has degree above 104, by a refereed result; above 108, by a claimed one. It has at least six preimages over a general point, and not a prime number of them if an unrefereed preprint holds. The two component degrees have a greatest common divisor of at least 18. The hatching is a different kind of wall. It rules out a mechanism, not a region: no counterexample in the plane has a torus symmetry, so none can be built the way every three-dimensional one was. Abhyankar adds that its components have exactly two points at infinity. Jung and van der Kulk add that every automorphism of the plane is a composition of affine and triangular maps, so a counterexample would be a Keller map outside that group.

Kraus stated the plane case in 1884. It is the oldest form of the problem and the only one left, and it now controls Dixmier's problem as well.

Questions

Is the symmetry forced? Alpöge's map, Gallagher's family, the factory maps and Harish's map all have a torus symmetry of mixed sign. If every counterexample must, then in the plane there are none, by Shaska's theorem, and JC₂ is true. The evidence is local. The ledger reports that at Alpöge's map and at Gallagher's degree-4 map, every first-order deformation of degree at most 7 that keeps the Keller condition is a composition or a symmetric one. That is question O5, and it is open.

Are the higher dimensions new? Any counterexample in ℂ³ becomes one in ℂ⁴ by adding a coordinate. Harish's map of ℂ⁴ has eight preimages, so it is not Alpöge's map padded. It might still be some other three-dimensional map padded, and no one knows how to tell. That is the stabilization question.

Some questions close as soon as they are written down beside the right fact. The ledger asked whether the automorphisms of ℂ³ and Alpöge's map generate every étale self-map. Degrees multiply under composition, so those generate only degrees that are powers of 3, and the ledger's own entry for Gallagher's degree-4 map answers no. Written beside each other, the entries answer question O1. The talk page asks whether any finite set of generators is enough.

How this page is made

Everything here is a wiki page, and every page can be edited, with its history kept and its author recorded. The relations are typed and each has a sentence that argues it: this implies that, this is a special case of that, this object refutes that statement. The network, the ladder and the charts are computed from those sentences, so correcting a sentence corrects every picture that depends on it. The help page has the syntax. Every family's cells, every wall, every tier and every date on this trail came from a research pass that opened its sources on 29 and 30 September 2026. Where a source could not be opened, the page says so.

Colophon

Written by Claude Opus 5.5 (claude-opus-5-5) in Claude Code 2.1.280, at the direction of Danielle Fong, on 30 September 2026. The research was drafted by parallel agents and checked by an adversarial second pass. That pass corrected 23 of the 31 entries it checked, for citations, relation types, provenance and a few overstatements. Several agents stalled, and the remaining six entries were checked in this session with direct web access. The atlas re-ran five certificates for this essay: Alpöge's map with two fibers, the master equation, Gallagher's degree-4 determinant, Harish's map, and Meng and Yang's potential. It did not rebuild any Lean or Isabelle project, did not re-run Gallagher's 98 verifiers or Thompson's, and did not check Long's Poisson map, Wilson's proof, Zheglov's proof or Moskowicz's proof. Each of those is marked on its page at the tier it has earned.