Vitushkin's construction method faces an obstruction in two dimensions that the three-dimensional polynomial counterexample circumvents
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
Vitushkin's 1999 example is the rational map (r, s) ↦ (r²s⁶ + 2rs², rs³ + 1/s), which has constant Jacobian −2 and sends (−3, −1) and (1, 1) to the same point. Its defining structural feature is a scaling symmetry: it is homogeneous for the weights (−4, 1), so both components are a polynomial in the single invariant w = rs⁴ divided by a power of s. The 2026 three-dimensional counterexample has exactly the same shape one dimension up: it is homogeneous for the mixed-sign weights (1, −1, −2), and closely parallels Vitushkin's construction lifted to three dimensions, down to the shared Jacobian of −2.
The obstruction the claim refers to is real and has a short proof. A July 2026 preprint by T. Shaska shows that every scaling-equivariant polynomial Keller map of the plane is a polynomial automorphism, whatever the signs of the weights: in the only case with room for a counterexample, the map descends to a one-variable polynomial whose derivative has degree exactly one less, and comparing degrees forces the components to be constants. A curve, in other words, has an exact ramification count that leaves the missing sheets nowhere to go. That is why Vitushkin's planar construction can only ever produce a rational map, with the pole along s = 0 doing the work that non-properness does in three dimensions. Over a surface the count no longer closes: the preimages that would have to merge can instead escape to infinity, and the announced map is the smallest instance in which that happens. The same point appears in the tangent-sweep account of the counterexample, where cancelling the Jacobian factor of a planar sweep requires padding by a third variable before twisting by monomial maps.
What the record does not settle is the historical attribution. Two informal sources say that Vitushkin himself observed the planar obstruction in 1999, but neither quotes him, and his paper could not be opened for this assessment; the preprint that proves the obstruction does not mention him. The claim as worded does not depend on that attribution, and it stands on the mathematics, which is elementary and was checked directly. Confirmation would come from a reading of Vitushkin's paper or from a refereed treatment connecting the graded-map theorem to his construction.
Full reasoning: the evidence and decisions behind this verdict
Direct verification. Using computer algebra, Vitushkin's map V(r,s) = (r²s⁶ + 2rs², rs³ + 1/s) was confirmed to have Jacobian determinant −2, to satisfy V(−3,−1) = V(1,1) = (3,2), and to be equivariant under (r,s) ↦ (t⁻⁴r, ts) with the components scaling by t⁻² and t⁻¹; equivalently V = ((w² + 2w)/s², (w + 1)/s) with w = rs⁴. The 2026 map (a, b, c) was confirmed to have Jacobian −2 and to scale by (t⁻², t⁻¹, t) under (x, y, z) ↦ (tx, t⁻¹y, t⁻²z). So both maps are graded constant-Jacobian maps of the same kind, and the mixed-sign grading of the 2026 map is settled by computation, as is the factual base of Vitushkin's 1999 rational example.
The obstruction. Theorem 3.4 of Shaska, "Graded Keller maps and the Jacobian Conjecture" (arxiv.org/abs/2607.20210, v2 of 25 July 2026), read in full, states that a Keller map of the plane equivariant for a nontrivial multiplicative-group action is a polynomial automorphism, and normal forms are given by weight type. The proof was followed step by step: linearizability of torus actions on the plane (Gutwirth 1962) reduces to a diagonal grading; the elliptic and parabolic cases give triangular or affine maps directly; in the hyperbolic case with coprime weights (p, −q) the invariant ring is C[u], u = x^q y^p, the map is (x f(u), y g(u)), det JG = fg + u(p f g' + q g f'), the descended map Ḡ(u) = u f^q g^p satisfies Ḡ' = f^{q−1} g^{p−1} det JG, and if det JG is a nonzero constant a degree comparison gives deg f + deg g = 0. Each identity was re-derived by hand and holds. This underwrites the planar equivariant-Keller theorem. The preprint's closing section ("Why dimension three") explains the mechanism: in dimension two the Keller condition descends to an identity between one-variable polynomials in which the Jacobian is a derivative, and the degree drop closes the case; in dimension three the quotient is a surface, the Jacobian is no longer a derivative, and missing sheets escape to infinity. Vitushkin's rational map escapes the theorem only because it is not polynomial: the corollary forcing target weights to be a permutation of source weights uses the Jacobian at the origin, where V has a pole.
Independent corroboration of the dimensional point. Gao, "Counterexamples to the Jacobian conjecture in dimensions greater than two" (arxiv.org/html/2608.00222), read through Section 3, describes the counterexample as a planar tangent sweep with Jacobian 2γ, which "can be cancelled by conjugation with monomial maps" only after one "pad[s] the sweep to three variables"; the construction "exists only in dimension ≥ 3" and the general framework is stated for n > 2. This is the same obstruction seen from the geometric side and does not depend on the grading formalism.
Instances and their weight. Three sources affirm; none denies. The Shaska preprint carries its own proof but does not mention Vitushkin, so it is recorded as an assertion of the claim with reduced confidence; the identification of its graded class with Vitushkin's method rests on the computation above rather than on any source. The Wikipedia talk-page comment (Sky19841, 29 July 2026) and an anonymous comment on Tao's blog (30 July 2026) both assert that Vitushkin himself identified the planar obstruction, without quotation; their wording is close enough that they may be one voice. The comment that started the parent dispute ("nothing new here ... in two dimensions one only gets a rational map") observes the rational-versus-polynomial difference but treats it as trivial; it neither asserts nor denies an obstruction and was not recorded as an instance of this claim. Vitushkin's paper (Mat. Zametki 66:2, 1999; Math. Notes 66, 245–249) could not be fetched: the mathnet.ru full-text request timed out and the zbMATH review returned a 403, so whether he stated the obstruction is unverified; the gist that circulates his example calls it a well-known "fake counterexample" but says nothing about an obstruction.
Weighing. The claim's mathematical content is established by direct checks and by a short proof that was read and re-derived; the only rigorous source is an unrefereed preprint days old, and the claim's phrase "Vitushkin's construction method" requires the interpretive step of taking the scaling-equivariant, monomially twisted shape as the method's defining feature. Those two reservations keep the status at supported rather than verified. What would move it: to verified, a refereed treatment or a reading of Vitushkin's paper confirming the identification; downward, a demonstration that the 2026 map's relation to Vitushkin's is superficial (which would make the claim ill-posed) or an error found in Theorem 3.4.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because Vitushkin's 1999 map is a rational, non-injective self-map of the plane with constant Jacobian that is equivariant for the scaling weights (−4, 1), and given that every scaling-equivariant polynomial Keller map of the plane is an automorphism, the same construction can never produce a polynomial counterexample in two variables: the denominator is forced. Since the 2026 map is homogeneous for the mixed-sign weights (1, −1, −2) and closely parallels Vitushkin's construction lifted to three dimensions, it is an instance of the same method in the first dimension where the degree count no longer closes, so the obstruction is real and the three-dimensional map gets past it.
The inference goes through: if the planar equivariant-Keller theorem holds and both maps are graded constant-Jacobian maps of the same kind, then the construction is blocked in two variables and unblocked in three exactly as the claim says. The weight rests on the theorem that every scaling-equivariant polynomial Keller map of the plane is an automorphism, whose short proof has been checked but which comes from an unrefereed preprint, and on the structural parallel between the 2026 map and Vitushkin's, which fixes what "the same method" means; the two computational premises are settled by direct verification.
Provenance
Where this claim has been said, linked to its canonical form.
The support rests on one primary document, a July 2026 preprint that proves no scaling-equivariant polynomial Keller map of the plane can fail to be an automorphism and explains why the mechanism behind the 2026 counterexample needs a third dimension; it does not mention Vitushkin. The other two sources are a Wikipedia talk-page comment and a blog comment, close in wording and possibly one voice, which restate the preprint's point as being about Vitushkin's method and add, without citation, that Vitushkin himself observed the obstruction in 1999. Vitushkin's paper could not be opened, so that attribution is unverified; the link between his construction and the preprint's theorem rests instead on the direct check that his map is equivariant for the weights (−4, 1). A reader should open the preprint's Theorem 3.4 and closing remarks first.
People concentrated on 2D because it’s the most attractive (with too many wrong proofs) and disproving it leads to disproving all dimensions. The latter is considered by Vitushkin, who clearly focused on finding the obstruction for an easy 2D counter example without caring about 3D or higher.
Replying to a comment that the counterexample contains "nothing new" beyond Vitushkin, the commenter calls that "a bit extreme" and characterises Vitushkin's 1999 work as having identified the obstruction to an easy two-dimensional counterexample, implicitly locating the dimensional difference as the substantive point.
Asserted without evidence of the source's own. A short anonymous comment with no argument or citation behind it. Its phrasing closely resembles the Wikipedia talk-page comment of the previous day and the two may be one voice.
In two cases a graded Keller map is a polynomial automorphism for a reason visible in the weights alone: elliptic weights in every dimension, where the orbits give properness, and dimension two for every signature, where the invariant ring is generated by one element and a degree count closes the argument. What remains is the hyperbolic case in dimension three and above, and Eq. 1 is an instance of it.
The preprint proves that a scaling-equivariant (graded) polynomial Keller map of the plane is always an automorphism, shows the 2026 counterexample is graded with mixed-sign weights in dimension three, and explains in its closing remarks why the mechanism needs dimension three (a curve has an exact ramification count that leaves no room). It does not mention Vitushkin by name; the link to Vitushkin's construction rests on his map being graded of the same kind.
The source's own evidence bears what it asserts. The preprint carries its own proof: a linearization theorem for torus actions on the plane reduces to a diagonal grading, and in the one case with room for a counterexample a degree count on the descended one-variable map forces the components to be constants. The closing remarks explain why dimension three is needed: a curve has an exact ramification count and the missing sheets have nowhere to go, whereas over a surface they can escape to infinity. Vitushkin is not named; the connection to his construction is that his rational map is equivariant for the weights (−4, 1) and so falls under the same graded framework. Worth reading closely: Theorem 3.4 and the "Why dimension three" remarks are the only rigorous statement of the planar obstruction on record; a later Steward should check whether the preprint has been refereed or revised, and whether any later version connects it explicitly to Vitushkin.
Inspired by the new counter example, a new preprint explained the key difference for this construction method between the 2D and 3D cases: https://arxiv.org/abs/2607.20210 . However, for 2D, the central obstruction to create counter examples using this method has already been observed in Vitushkin's "Evaluation of the Jacobian of a rational transformation of C^2 and some applications" in 1999. This makes the difference of dimensions very important.
In a discussion of how to categorise the article after the three-dimensional counterexample, an editor argues the dimensional difference is essential, citing the Shaska preprint for the 2D/3D difference in the construction method and attributing the 2D obstruction to Vitushkin's 1999 paper.
The assertion outruns the source's own evidence. The comment rests on two supports of different quality. The preprint it cites does prove that the graded construction cannot yield a polynomial counterexample in the plane and does explain why dimension three is needed, though it never mentions Vitushkin. The further statement that Vitushkin himself observed the central obstruction in 1999 is offered without quotation or page reference, and could not be checked against Vitushkin's paper, whose text was not reachable.
How these sources relate
- https://en.wikipedia.org/wiki/Talk:Jacobian_conjecture draws its statement from https://arxiv.org/html/2607.20210v2, stating it more strongly than that document supports. The comment takes the 2D/3D distinction from the preprint and states it as a fact about "this construction method", meaning Vitushkin's. The preprint does prove the planar result and explain why dimension three is needed, but it frames the class as graded Keller maps and never names Vitushkin or his construction; that identification, and the further attribution of the obstruction to Vitushkin's own paper, are the commenter's additions.
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.