jacobian manuscript

Minor revision. Both independent referees independently recomputed the central Jacobian identity and the three-point fiber, found no mathematical error, and recommend the same disposition. The manuscript exhibits an explicit polynomial map F : C^3 → C^3 with det Jac F ≡ −2 and three distinct rational points mapping to a common image, thereby falsifying the Jacobian conjecture in dimension three and, by product stabilization, in every dimension n ≥ 3. That claim, if accepted after the presentational repairs below, is decisive for affine algebraic geometry.

What is right about the paper is unusually clear. The counterexample is fully explicit, with rational coefficients and rational witness points, so it works over every field of characteristic zero. The reduction of fiber geometry to simple roots of a binary cubic via the projective coordinate [x : 1+xy] is geometrically transparent and determines every fiber, the image, and the nonproperness set in one stroke. Theorem 2.2 cleanly separates the false global assertion (“every Keller map is proper”) from the true refined statement (“every proper Keller map is a polynomial automorphism”). The final section constructs parametric families realizing every generic degree d ≥ 3, showing the phenomenon is robust rather than accidental. The exposition is short, dense, and essentially free of superfluous material.

What remains load-bearing is presentational integrity rather than mathematical gap. The central claim rests on det Jac F ≡ −2 together with the three-point fiber at (−1/4, 0, 0). Both referees verified the chain-rule computation through the coordinate change (P, y, s) with s = x/(1+xy), obtaining Jacobian factors −A and 2/A whose product is −2 on the dense open set A ≠ 0, with the identity extending by polynomial continuity. The three rational points (0,0,−1/4), (1,−3/2,13/2) and (−1,3/2,13/2) all map to (−1/4,0,0) by direct substitution. Because the claim settles a classical open problem, the published version must contain either an expanded hand computation of the two Jacobian matrices or a short, self-contained computer-algebra transcript (Macaulay2, Singular, or Sage) that independently expands det Jac F and confirms the constant −2. This is a presentational requirement, not a mathematical objection; the existing argument is already rigorous.

Per-section diagnostics are as follows. Section 1 correctly credits the announcement and isolates the surviving positive statement. Section 2 proves the properness refinement and the codimension-at-least-two statement for the complement of the image; the covering-map argument is standard but compressed, and a parenthetical textbook reference would help non-specialists. Section 3 carries the explicit map and the determinant identity; this is the load-bearing computation and needs the expanded verification artifact. Section 4 determines image, nonproperness set Σ (the discriminant hypersurface), and the nonreduced singular scheme along the smooth rational curve Γ; a one-line remark that the ordinary cubic discriminant is 4Δ would aid comparison with classical tables. Section 5 constructs the degree-d families; the proof should explicitly record that the added η terms vanish at the original three-point collision when η_{d−2}(T) = c(4T+1), so the same fiber witness survives.

The explicit revision contract is therefore: (1) supply an expanded hand computation or a self-contained computer-algebra script verifying det Jac F ≡ −2 and the three coincident images; (2) correct the three bibliographic entries that fail resolver checks - Bass-Connell-Wright DOI broken by an embedded space, Keller 1939 DOI capitalization/status, and Stacks Project Tag 0F2P bibliographic format; (3) fix the abstract typo “deteminant,” the spacing artefacts in section headings (“Beha vior,” “F amilies”), and the “anétale” spacing error; (4) add a brief parenthetical reference for the proper-local-homeomorphism-is-covering step; (5) note explicitly in §5 that the deformation preserves the three-point collision because η(−1/4) = 0. Optional but recommended: display the refined positive statement as a numbered corollary for future citation, and state the use of Chevalley’s constructibility theorem when arguing that the complement of the image has codimension at least two.

What would alter the decision is a mathematical error in the Jacobian identity or in the three-point fiber. None has been found by either referee. If an independent computer-algebra check produced a non-constant determinant, the recommendation would move immediately to reject. Conversely, if the authors supply the verification script and repair the bibliography, the paper is ready for acceptance at a top algebra or algebraic-geometry venue. Relative to field standards, the manuscript already exceeds the usual bar for an explicit counterexample paper: the geometry is fully determined, not merely a single bad fiber, and the families show the mechanism is structural. The two-dimensional Jacobian conjecture remains untouched, and the paper correctly emphasizes that a codimension-two complement of the image is insufficient for properness.

Disclosures: a third referee seat was unavailable (output truncated), so this decision rests on two high-confidence independent reports that agree on both mathematics and disposition. No formal-canon overlap exists; the paper is pure affine algebraic geometry over characteristic zero. The reference-integrity audit flagged three suspect entries that must be repaired before publication. No concrete AI-output artifact (hallucinated reference content beyond malformed DOIs, placeholder text, or meta-commentary) was found in the mathematical body.

The manuscript’s central construction is the polynomial map F : C^3 → C^3 announced by Alpöge. After the coordinate change to (P, y, s) with s = x/(1+xy) and A = 1+xy, the authors compute the Jacobian matrix of the transformed map and obtain diagonal-block factors whose determinants multiply to −2 on the dense open set A ≠ 0. Because det Jac F is itself a polynomial, the identity det Jac F ≡ −2 extends from the dense open set to all of C^3 by continuity (or, equivalently, by the identity theorem for polynomials). Both referees recomputed this chain-rule argument and confirmed the constant −2. The same section exhibits three explicit rational points (0,0,−1/4), (1,−3/2,13/2), (−1,3/2,13/2) that all map to (−1/4,0,0). Direct substitution verifies the common image, so F is a Keller map that is not injective and therefore not a polynomial automorphism. Stabilization by product with the identity on C^{n−3} immediately yields counterexamples in every dimension n ≥ 3.

Section 2 isolates the surviving positive statement. Proposition 2.1 records that the complement of the image of any Keller map has codimension at least two; the argument uses that a Keller map is étale (hence open) and that the complement is constructible, so a codimension-one component would contain an irreducible divisor, contradicting dominance of an étale map of smooth varieties of equal dimension. Theorem 2.2 then proves the equivalence, for any Keller map G, of the four conditions: G is a polynomial automorphism; G is proper; the nonproperness set S_G is empty; and codim S_G ≥ 2. The key step is that a proper local biholomorphism with finite fibers is a covering map, and the only covering of the simply-connected space C^n is one-sheeted, so G is bijective and the algebraic inverse is regular by the usual Jacobian criterion. The argument is standard but compressed; a parenthetical reference to Forster or Huybrechts would assist readers outside several complex variables.

Section 3 introduces the projective coordinate [x : 1+xy] and reduces the fiber problem to a binary cubic in two homogeneous variables. Simple roots of that cubic are exactly the affine preimages. This description is the geometric heart of the paper: it determines every fiber (cardinality equal to the number of simple roots), the image (complement the smooth rational curve Γ cut out by the vanishing of the cubic’s discriminant locus in a suitable sense), and the nonproperness set (the discriminant hypersurface Σ). The reduction is elementary and fully explicit; no resolution of singularities or heavy commutative algebra is required.

Section 4 studies the behavior at infinity. The normalized discriminant Δ is introduced, and the singular scheme of Σ is computed ideal-theoretically; it is nonreduced along Γ. A one-line remark that the ordinary cubic discriminant equals 4Δ would help readers comparing with classical invariant-theory tables. The nonproperness set is identified with Σ, confirming that S_F is a hypersurface (codimension one) and therefore that F fails every equivalent condition of Theorem 2.2, as expected for a counterexample.

Section 5 constructs families. Starting from the seed map F, the authors add terms involving a univariate polynomial η_{d−2} of degree d−2, chosen so that the Jacobian determinant remains the nonzero constant −2 (or is renormalized to 1) while the generic degree becomes d. The same three-point collision survives whenever η_{d−2}(T) vanishes at T = −1/4, for instance when η_{d−2}(T) = c(4T+1). This produces nonproper Keller maps of every generic degree d ≥ 3. The construction is algebraic and works over any field of characteristic zero. The proof should explicitly record that the added terms in the second and third coordinates vanish at the three source points because B ≠ 0 there and η(P) = 0 at P = −1/4.

Domain of validity is characteristic zero (the binary-cubic analysis and the covering-space step both use char 0). No counterexample candidate inside the paper’s own hypotheses was found: the three-point fiber is explicit, the determinant identity is polynomial, and the properness equivalence is classical once properness is assumed. Comparison with prior literature is straightforward. The Jacobian conjecture has been open since Keller (1939); the Bass-Connell-Wright reduction to degree ≤ 3 is cited; the present work does not touch dimension two and does not claim to. The geometric analysis goes well beyond a single bad fiber: image, nonproperness locus, and degree-d families are all determined. Relative to the formal canon supplied for this review, there is no overlap; the paper lives entirely in classical affine algebraic geometry.

If correct, the result settles one of the oldest open problems in affine algebraic geometry negatively in all dimensions greater than two. The two-dimensional case remains open, and the paper cleanly isolates the surviving positive statement that every proper Keller map is a polynomial automorphism. The geometric analysis and the degree-d families supply a flexible source of further counterexamples and make the mechanism transparent rather than accidental. The methods are elementary and explicit, which both strengthens confidence and makes independent verification straightforward.

• Fully explicit counterexample with rational coefficients and rational witness points; works over every field of characteristic zero.
• Binary-cubic reduction of fibers is geometrically transparent and determines image, nonproperness set, and generic degree in one stroke.
• Theorem 2.2 cleanly separates the false global statement (“every Keller map is proper”) from the true refined statement (“every proper Keller map is an automorphism”).
• Parametric families produce nonproper Keller maps of every generic degree ≥ 3, showing the phenomenon is robust.
• Short, dense, and carefully written; essentially no superfluous material.
• Honest scoping: dimension two is explicitly left open; codimension-two image complement is correctly distinguished from properness.
• Both available referees independently recomputed the load-bearing identities and found no mathematical error.