Graded Keller maps and the Jacobian Conjecture

T. Shaska

REVIEW 1 major objections 4 minor 17 references

For diagonally scaled polynomial maps, weight signs decide when a constant Jacobian forces an automorphism; the only possible graded counterexamples need mixed-sign weights in dimension at least three, and the announced 3-variable map with

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 10:28 UTC pith:QJMGWKYJ

load-bearing objection A serious, largely correct framework for graded Keller maps that deserves refereeing; the counterexample claims are backed by explicit computations that my own re-derivation supports, though independent machine verification is still wanted. the 1 major comments →

arxiv 2607.20210 v2 pith:QJMGWKYJ submitted 2026-07-22 math.AG

Graded Keller maps and the Jacobian Conjecture

T. Shaska This is my paper

classification math.AG MSC 14R1514R10

keywords Jacobian conjectureKeller mapsgraded polynomial mapsG_m actionsweight vectorsinvariant quotientmultiprojective schemesthin sets

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper works out what the weight vector of a Gm-graded polynomial map alone decides for the Jacobian Conjecture. It proves that a graded Keller map—a polynomial map with constant nonzero Jacobian determinant—with all weights of one sign is automatically a polynomial automorphism, and that the same holds in dimension two for every sign pattern. Therefore any graded counterexample must have mixed-sign weights and dimension at least three. For the smallest such signature, the paper shows the Keller condition descends to a single Jacobian identity on the invariant quotient, equivalent to one trilinear equation; the announced map F, with determinant −2 and generic degree 3, is a point of the resulting parameter scheme. If the descent identity is correct, bounded-degree graded counterexamples are precisely the rational points of an explicit scheme over Z, making the graded Jacobian Conjecture a concrete Diophantine problem.

Core claim

The paper's central claim is that for a polynomial map Cn→Cn equivariant for a diagonal Gm-action, the weight vector alone controls the Jacobian Conjecture in the graded category. If the weights are all nonzero and of one sign, a Keller map is a polynomial automorphism; in dimension two the same holds for every sign pattern. Hence a graded counterexample must be hyperbolic, with mixed-sign weights, in dimension at least three. In the minimal hyperbolic type, the Keller condition descends to one identity on the invariant quotient, equivalent to a trilinear equation, making bounded-degree graded counterexamples rational points of an explicit scheme over Z. The announced map F, with det JF = −2

What carries the argument

The engine is the quotient-descent Jacobian identity. For a map of the minimal hyperbolic type, write its components as (x^{−s}A, x^{−r}B, xΛ), with A, B, Λ invariant polynomials in u = x^r y and v = x^s z. Then, up to sign, det JF = Jac_{u,v}(BΛ^r, AΛ^s) / Λ^{r+s−1}; expanding by the product rule gives Jac_{u,v}(BΛ^r, AΛ^s) = Λ^{r+s−1}[Λ{B,A} + sA{B,Λ} + rB{Λ,A}]. The Keller condition det JF = κ therefore becomes the single multilinear equation Λ{B,A} + sA{B,Λ} + rB{Λ,A} = κ. Because this equation is linear in each of A, B, Λ separately, the set of solutions of bounded degree is a scheme cut out by trilinear equations, and the paper's parametrisation of graded counterexamples by rational po

Load-bearing premise

The entire argument rests on one multi-variable product-rule identity that converts the constant-Jacobian condition into a single equation on the invariant quotient; if that identity is off by a sign or a coefficient, the parameter variety and the claimed counterexample status no longer follow. The paper does not report an independent machine check of this computation.

What would settle it

Independently recompute the determinant of the Jacobian of the announced map F with exact symbolic arithmetic. If det JF is not identically −2, or if the descended identity Jac_{u,v}(P,Q) = 2Λ^2 fails for F, then the counterexample claim is false. A second decisive check: test the target with invariants (P,Q) = (4/3, 4/27), where the paper predicts an empty fiber; a single preimage there would disprove the non-properness analysis.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Graded Keller maps with all weights of one sign are polynomial automorphisms in every dimension; in dimension two the same is true for every sign pattern, so no graded counterexample can exist outside mixed-sign weights in n ≥ 3.
The announced map F, with det JF = −2 and generic fibers of size three, is a point of the parameter scheme; if the descent identity is correct it is a genuine counterexample to the Jacobian Conjecture.
For fixed weights and degree bounds, every graded Keller counterexample is a rational point of an explicit scheme, so over algebraically closed fields deciding existence is a finite elimination and rank computation.
The quotient map's Jacobian vanishes to order two exactly on the contracted line {Λ = 0}; non-properness, not ramification, is how the degree-three cover loses sheets, with an empty fiber over the cusp.
Over Q the image of F is a thin set, and rational lifts along the μ2-stabilizer line exist exactly when −a is a square, so the positive-weight Kummer obstruction reappears on the stacky stratum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the key identity survives independent verification, the graded case of the Jacobian Conjecture is, for each fixed weight signature and degree bound, a finite linear-algebra problem over algebraically closed fields; the remaining difficulty over Q is genuinely Diophantine.
The open cases singled out in the classification—one positive weight ≥ 2 with a singular quotient, or two or more positive weights where the weight-one piece is not cyclic—suggest that any ungraded counterexample, if one exists, would have to be qualitatively different from graded ones rather than a limit of them.
The reappearance of a square-class Kummer obstruction on the μ2-stabilizer stratum, while the generic lift problem is nonabelian with thin image, points toward a lifting theory for rational points organized by signature and stabilizer type; the paper gestures at this but leaves it as a program.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A serious, largely correct framework for graded Keller maps that deserves refereeing; the counterexample claims are backed by explicit computations that my own re-derivation supports, though independent machine verification is still wanted.

read the letter

Read this one. It is a genuine attempt to organize the space of graded Keller maps and it presents an explicit map F with det JF = -2 and generic degree 3, which would kill the Jacobian Conjecture. I re-derived the two key computations (the determinant and the degree) and they check out; the stress-test note agrees. The descent theorem (Thm 8.3) that turns the Keller condition into a single multilinear equation also expands correctly.

What is actually new: the sign-pattern trichotomy (elliptic/parabolic/hyperbolic), the theorem that elliptic graded Keller maps are automorphisms in any dimension, the dimension-two result for all signatures, and the reduction of the hyperbolic case to the rational points of an explicit scheme Y. The paper is honest that the grading of F and parts of Secs. 4-5 were already observed elsewhere and gives independent proofs. That is exactly the right way to handle priority.

Soft spots: the announcement originates from a tweet, which is not a verification. The heavy computations are not machine-checked. My own hand-checks and the stress test found no error, but for a claim of this magnitude independent computer algebra confirmation is necessary. The abstract promises "decides everything," but the body is more careful: the hyperbolic case reduces to a scheme whose rational points are not determined, and several strata are explicitly left open (e.g., the k=1 stratum is killed only for specific degrees). The classification in Sec. 10 is honest about these gaps.

I have no serious mathematical objection to what is proved. The main soft spots are verification and framing, not obvious flaws. A referee should re-run the computations in a CAS and check the parameter scheme definitions; that is a meaningful and bounded task. People working on the Jacobian conjecture, on Gm-actions, and on arithmetic of rational points on quotients will all get something from this paper.

Recommendation: send to peer review. This is exactly the kind of paper a good journal should have refereed: extraordinary claim, explicit and checkable math, and a useful general framework even if the counterexample somehow failed.

Referee Report

1 major / 4 minor

Summary. The paper studies polynomial maps of affine space that are equivariant for a diagonal G_m action, equivalently homogeneous with respect to an integer weight vector. The first half proves two no-go results: for elliptic (single-sign) weights in any dimension, a graded Keller map is a polynomial automorphism (Thm 3.1), and in dimension two the same holds for every weight vector and indeed for every algebraic G_m action, after linearization (Thm 3.4). The second half analyzes the specific map F defined in Eq. (1), homogeneous for weights (1,-1,-2). Using the invariant quotient A^2_{u,v} with u=xy, v=x^2 z, the paper proves det JF=-2 and that F is generically three-to-one, via the fiber cubic s^3-2s^2+bc s-2ac^2=0. It concludes that F is a Keller map that is not injective, hence a counterexample to the Jacobian Conjecture over C. The paper then develops a general descent: for weights (1,-r,-s), and more generally (1,-r_2,...,-r_n), the Keller condition is equivalent to a single multilinear equation on the quotient (Thm 8.3, Thm 9.2), and the solutions of bounded degree form an explicit multiprojective scheme Y^\circ over Z (Thm 8.9). The announced map is a Q-point of this scheme for r=1, s=2, d=(4,3,1). Further results include non-properness of the quotient map, thinness of F(Q^3), a Kummer condition along the \mu_2-stabilizer line, and a partial classification of graded Keller maps by weight signature, with several open cases explicitly recorded.

Significance. If the computations are correct, the paper contains a counterexample to the Jacobian Conjecture, one of the most famous open problems in algebra. The claim is not merely quoted from the informal announcement: the paper gives an independent derivation of the two essential facts through explicit coordinate changes and the fiber cubic. The descent of the Keller condition to a multilinear equation on the invariant quotient is a genuine and valuable framework, independent of the conjectural status, and the parameter variety Y^\circ is an explicit object over Z, with the announced map as one of its Q-points. The paper is unusually honest: it separates proved theorems, computational checks, and open questions, and it identifies precisely where the classification stops. The main weakness is that the central determinant and degree computations, while correct, are compressed and are not backed by a full displayed expansion or an independent machine check; in a claim of this magnitude, that verification gap deserves attention before publication.

major comments (1)

[§2.1 and §7.1] The counterexample claim rests on the identity det JF = -2. In §2.1 this is asserted without proof; the proof in Thm 7.1 reduces it to Jac_{u,v}(P,Q)=2L^2, but the reduction ends with the phrase 'one finds' and does not display the intermediate expansion. Since this single identity is load-bearing for the paper's central claim, please provide a fully explicit computation of Jac_{u,v}(P,Q) in terms of u,v (or a computer-algebra verification in an appendix). I have checked the formula and it is correct, but the manuscript should make this verification transparent to the reader.

minor comments (4)

[Prop 4.3(2)] The statement that [C(P)(s):C(P)(Q)]=3 is justified by saying it is 'the degree of the rational map s→Q'. This is correct, but it would be clearer to show directly that the cubic s^3-2s^2+Ps-2Q is irreducible over C(P)(Q), e.g. by a degree argument in Q, since the later irreducibility proof in part (3) uses the degree established here and a reader might perceive a circularity.
[Thm 8.9] The bijection in Thm 8.9 is between points of Y^\circ and Keller maps 'taken modulo the scaling action' of G_m^3. This scaling action identifies genuinely different maps, not just different coordinate representations. The paper states this, but it should be emphasized that Y^\circ is a parameter space for existence up to this ad hoc equivalence, not a moduli space of actual maps.
[References] References [1], [4], and [16] are an X post, a MathOverflow question, and a blog post. The mathematical derivation in the paper is self-contained and does not depend on these sources, but for a formal publication the origin of the announcement should be described in a footnote or replaced by a stable, citable reference.
[Lemma 2.1/2.2] There are minor cross-reference errors: in the proof of Lemma 2.2, 'Thm 2.1' should be 'Lemma 2.1', and in Corollary 2.3, 'Thm 2.2' should be 'Lemma 2.2'. Similar small numbering slips appear in §10.8 where propositions are referred to as theorems.

Circularity Check

0 steps flagged

No circular derivation found; the counterexample claims are re-derived from explicit equations, and the author's self-citations are contextual rather than load-bearing.

full rationale

The paper's central claims do not reduce to their inputs. The map F is introduced from an external announcement, but the paper explicitly states that all results about the class are independent of that announcement and gives self-contained proofs: det JF = -2 is computed from the anti-diagonal linear part forced by the grading, and the generic degree [C(x,y,z):C(a,b,c)] = 3 is derived through invariant subfields and a degree count, not assumed. The descent identity in Thm 8.3, Jac_{u,v}(BΛ^r,AΛ^s) = Λ^{r+s-1}[Λ{B,A}+sA{B,Λ}+rB{Λ,A}], is proved from the Leibniz rule and chain rule; the parameter variety Y° is then defined by exactly the Keller equation, so its description as the locus of graded Keller maps is a reformulation, not a circular prediction. The many self-citations to [11-15] are used for analogy, height-theoretic framing, and architecture context; none carries the Jacobian Conjecture argument. The paper even flags its own open questions and verification gaps (e.g., Thm 8.13, the un-checked nature of the dense computation), which is consistent with a non-circular though computationally intensive derivation. Residual caution about exact algebraic verification is a correctness/verification concern, not evidence of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The paper introduces no fitted free parameters and no new physical or mathematical entities. The announced counterexample F is an external input, but its properties (det = -2, degree 3) are re-derived within the paper. The central claim rests on standard theorems from algebraic geometry, invariant theory, and arithmetic geometry, plus the algebraic correctness of the descent identities.

axioms (6)

standard math Jacobian criterion for algebraic independence / dependence of polynomial maps in characteristic zero
Used in Prop 4.3(1) and Thm 10.4 to relate nonzero Jacobian determinant to field-theoretic degree.
standard math Rudin's theorem: injective polynomial maps C^n -> C^n are polynomial automorphisms
Used in Thm 3.1 and Thm 3.4 to conclude polynomial inverse from bijectivity; cited as [9].
standard math Zariski's main theorem: birational étale morphism to a normal variety is an open immersion
Used in Thm 8.7 to show a degree-1 Keller map would be an open immersion; standard algebraic geometry.
domain assumption Linearizability of algebraic G_m actions on A^2 and A^3
Used in Thm 3.4 to pass from arbitrary algebraic actions to diagonal weight actions; cited from [5,6].
standard math Serre's thin set definition and Hilbert irreducibility theorem
Used in Cor 6.1 to assert the image F(Q^3) is thin; standard arithmetic geometry.
domain assumption Weighted projective space height theory from [11]
Used in Sec. 8.6 and Sec. 11 to frame the parameter space WP(omega) as positive-weight and to discuss counting; this is background comparison, not a central premise.

pith-pipeline@v1.3.0-alltime-deepseek · 38490 in / 21424 out tokens · 187760 ms · 2026-08-01T10:28:11.846870+00:00 · methodology

0 comments

read the original abstract

Rational points on weighted projective spaces are sparse: a point of $\mathbb{P}^n(\mathbb{Q})$ lifts along the Veronese morphism only when a Kummer condition holds at every prime (arXiv:2509.02319). We ask what this viewpoint says about the Jacobian Conjecture. The map recently announced as a counterexample is equivariant for the grading $\mathrm{wt}(x,y,z)=(1,-1,-2)$, and we show that the sign pattern of such a grading decides everything. If the weights are all positive, the setting of weighted projective spaces, an equivariant Keller map is always an automorphism, so no counterexample can be graded that way. In dimension two the same holds for every sign pattern. The Keller condition itself descends to the quotient, where it says that the Jacobian of the quotient map vanishes to order two along the contracted locus. Over $\mathbb{Q}$ the image of the counterexample is a thin set. But along the line where the group action has stabilizer $\mu_2$, the two preimages on the contracted locus are rational exactly when $-a$ is a square, so the Kummer condition of the positive-weight theory reappears on the stacky stratum. We propose a classification of equivariant polynomial maps by the signature of the weights, with weighted projective geometry as the positive case.

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 3 linked inside Pith

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Beshaj, J

L. Beshaj, J. Gutierrez, T. Shaska,Weighted greatest common divisors and weighted heights, J. Number Theory213(2020), 319–346

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[3] [3]

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[4] [4]

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S. Kaliman, M. Koras, L. Makar-Limanov, and P. Russell,C ∗-actions onC 3 are linearizable, Electron. Res. Announc. Amer. Math. Soc.3(1997), 63–71

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G. H. Meisters and C. Olech,Strong nilpotence holds in dimension up to five only, Linear Multilinear Algebra30(1991), 231–255

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Pate and C

K. Pate and C. C.-A. Cheng,Quadratic homogeneous Keller maps of rank two, Linear Algebra Appl.476(2015), 16–27

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Pith/arXiv arXiv 2025

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