Counterexamples to the Jacobian conjecture in dimensions greater than two

Technical Summary: Counterexamples to the Jacobian Conjecture in Dimensions Greater Than Two

Problem Statement
The Jacobian conjecture, posed by Keller in 1939, asserts that any polynomial map $F: \mathbb{C}^n \to \mathbb{C}^n$ with a constant nonzero Jacobian determinant (a Keller map) must be a polynomial automorphism (i.e., possess a polynomial inverse). While the conjecture remains open for $n=2$, this paper addresses the status of the conjecture for dimensions $n > 2$. The work builds upon the refutation of the conjecture in dimension three announced by Alp¨oge (July 19, 2026), which provided a specific counterexample, and Gallagher's subsequent construction of an infinite family of counterexamples. The central problem addressed here is to generalize the mechanism behind these counterexamples to all dimensions $n > 2$ and to construct explicit examples with arbitrarily large geometric degrees.

Methodology
The paper employs a geometric construction based on the "tangent sweep" mechanism, originally isolated by Speyer. The methodology proceeds in three main stages:

1. Tangent Sweeping: The author constructs a polynomial map by sweeping a tangent direction field over a parametrized hypersurface. For a parametrized hypersurface $X: \mathbb{C}^{n-2} \to \mathbb{C}^{n-1}$ and a polynomial tangent field $\Delta$, a "padded sweep" map $F_0$ is defined. The Jacobian determinant of this sweep is shown to be a polynomial in a multiplier variable $\gamma$, specifically of the form $\det J(F_0) = \sum L_k \gamma^{k+1}$.
2. Branch Selection and Normalization: The construction relies on selecting specific "branches" where the Jacobian determinant simplifies to a monomial term $c \gamma^{k+1}$. This requires satisfying tangency criteria ($\det(\Delta, J(X)) \equiv 0$) and normalization conditions where specific coefficients $L_k$ vanish or become constants. The paper analyzes different types of direction fields (curve-like, tautological, and mixed) to determine which branches are available in various dimensions.
3. Monomial Twist: To convert the sweep (which has a non-constant Jacobian) into a Keller map (constant Jacobian), the author applies a "monomial twist." This involves composing the sweep with monomial maps and an affine stage to cancel the $\gamma$ factor in the Jacobian. This process introduces "side conditions" (divisibility constraints) on the coefficients of the polynomials defining the hypersurface. Crucially, this twist pushes the ramification locus (where the map fails to be injective) to infinity, resulting in a map that is everywhere unramified (étale) but not proper.

Key Contributions and Results

• General Framework: The paper establishes a general construction valid for every dimension $n > 2$. It demonstrates that by sweeping tangent direction fields on parametrized hypersurfaces and applying the monomial twist, one can generate Keller counterexamples in any dimension $n \ge 3$.
• Arbitrary Geometric Degree: The construction produces counterexamples of arbitrarily large geometric degree (the number of points in a generic fiber). For $n=3$, this recovers Gallagher's family. For $n \ge 4$, the curve-type direction field allows the geometric degree to be increased arbitrarily by choosing data of high degree.
• Five Explicit Counterexamples: The author works out five new explicit polynomial maps:
  • Dimension 3: A map $G$ with component degrees 4, 11, and 12 (geometric degree 4), based on a rational quartic curve with two cusps and a node.
  • Dimension 4:
    • $F_4$: A map with component degrees 4, 11, 12, and 21 (geometric degree 5) derived from a non-cylindrical ruled surface.
    • $F_5$: A map with component degrees 3, 12, 14, and 16 (geometric degree 10) derived from the "M-branch" of a mixed direction field.
  • Dimension 5:
    • $F_6$: A map with component degrees 7, 38, 40, 42, and 44 (geometric degree 6) derived from the "bottom branch" of a mixed direction field $\Delta = (1, w_1, w_1^2, w_2)^T$.
    • $F_7$: A map with component degrees 7, 86, 89, 92, and 95 (geometric degree 12) derived from the "curve-type" direction field, utilizing transverse Keller-type data.
• Fiber Structure Analysis: The paper provides detailed descriptions of the fiber structures for these maps. It proves that the generic fiber size equals the geometric degree, while specific fibers degenerate to smaller sizes (or become empty) over specific loci (e.g., the Jelonek hypersurface). For instance, the map $G$ attains every fiber size in $\{4, 3, 2, 1, 0\}$.
• Rigidity Phenomenon: In dimension 5, the analysis of the "middle branch" for the mixed direction field reveals a rigidity phenomenon. The governing equation for this branch degenerates into a linear transport equation on a characteristic-invariance locus, suggesting that genuinely five-dimensional data cannot realize this branch; solutions appear to be suspensions of four-dimensional sweeps.

Significance and Claims
The paper claims to provide a self-contained account of the tangent-sweep mechanism, generalizing it from plane curves to direction fields on hypersurfaces. Its primary significance lies in:

1. Completing the Refutation: It confirms that the Jacobian conjecture is false for all dimensions $n > 2$, providing a unified mechanism for counterexamples across all such dimensions.
2. Explicitness: Unlike previous abstract existence proofs, this work provides five fully explicit polynomial maps with verified Jacobian determinants and fiber structures.
3. Geometric Insight: It clarifies the geometric nature of the counterexamples: they are étale coverings $\mathbb{C}^n \to \mathbb{C}^n$ that fail to be injective solely because distinct preimages escape to infinity rather than colliding at finite points.
4. Propagation: The construction allows for the propagation of counterexamples along the sequence $n \mapsto n+3$, suggesting that counterexamples in lower dimensions can serve as data for constructing counterexamples in higher dimensions.

The author notes that all polynomial identities and fiber counts were verified using exact rational arithmetic and Gröbner bases where feasible. They acknowledge that the complete stratification of fibers for the higher-dimensional maps ($F_4$ through $F_7$) remains a subject for future work, though the generic fiber counts are rigorously established.