Deciding the component dichotomy for the Jacobian-one varieties $V_n^k$: automorphism versus counterexample components and the two-dimensional Jacobian conjecture

Xavier (June 2026), and independently Jelonek (July 2026), proved that every irreducible component $\Omega$ of the variety $V_n^k$ of polynomial maps $F:\mathbb{C}^n\to\mathbb{C}^n$ with $\deg F\le k$ and $\det DF\equiv1$ either consists entirely of polynomial automorphisms, as the Jacobian conjecture claims, or has as its general element a Keller counterexample - but neither work determines which alternative holds for any specific component. One month later the picture changed decisively: Alpöge (19 July 2026) announced an explicit three-dimensional Keller counterexample, Gallagher produced an infinite family with $\det DF\equiv1$, Speyer and Tao explained the tangent-sweep mechanism, and Gao generalized it to every dimension above two; the Jacobian conjecture is thus false for all $n\ge3$, and the existence question left open by Xavier is answered - the components containing the known counterexamples are generically counterexample components. The open problem is the determination that remains: decide, for every $n\ge2$ and $k$ and every irreducible component of $V_n^k$, which dichotomy alternative holds. In dimension two the determination settles the still-open two-dimensional Jacobian conjecture; in dimensions $n\ge3$ it is the classification of automorphism components versus counterexample components, including the irreducibility of $X(n,d)$, where Jelonek showed that an irreducible $X(n,d)$ with $n\ge3$, $d\ge6$ has generic element a counterexample.

Background

The Jacobian conjecture (JC), raised by Keller in 1939, concerns polynomial maps $F=(F_1,\dots,F_n):\mathbb{C}^n\to\mathbb{C}^n$. If $F$ has a polynomial inverse, the chain rule forces its Jacobian determinant $\det DF$ (the determinant of the matrix of first partial derivatives) to be a nonzero constant. A polynomial map with $\det DF$ a nonzero constant is called a Keller map, and JC asserts the converse: every Keller map is a polynomial automorphism, that is, has a polynomial inverse. The conjecture is easy in one variable (a univariate polynomial with constant nonzero derivative is linear), was verified for maps of degree at most two by Wang, and was popularized by the survey of Bass, Connell and Wright, whose reductions (degree reduction and formal expansion of the inverse) shaped most later attacks. Until July 2026 it was open in every dimension $n\ge2$.

Xavier's organizing principle fixes integers $n,k$ and studies the problem one component at a time. Let $V_n^k$ be the algebraic set of Keller maps $F:\mathbb{C}^n\to\mathbb{C}^n$ with $\deg F\le k$ and $\det DF\equiv1$, viewed as a subset of the coefficient space; write $\mathrm{Aut}(\mathbb{C}^n)$ for the group of polynomial automorphisms. JC is the claim $\bigcup_{n,k}V_n^k\subset\mathrm{Aut}(\mathbb{C}^n)$. An irreducible algebraic set is one that cannot be written as a union of two proper algebraic subsets, an irreducible component is a maximal irreducible algebraic subset, and 'the general $F\in\Omega$' means all $F$ outside a proper Zariski-closed subset (a proper subset cut out by polynomial equations in the coefficients) of $\Omega$. Xavier's Theorem 1.1 states a dichotomy: either $\Omega\subset\mathrm{Aut}(\mathbb{C}^n)$ or the general $F\in\Omega$ is not an automorphism. The proof shows that the invertible elements $I_n^k=V_n^k\cap\mathrm{Aut}(\mathbb{C}^n)$ form a Zariski-closed subset of $V_n^k$; the ingredients are Chevalley's theorem that projections of algebraic sets are constructible (finite Boolean combinations of algebraic sets), the Ax-Grothendieck theorem that every injective polynomial self-map of $\mathbb{C}^n$ is bijective, and a topological-degree argument showing closedness under locally uniform limits. Via Ax-Grothendieck, Xavier also derived a probabilistic criterion: almost surely, the first alternative holds for $\Omega$ if and only if, for a random choice of $(F,c)\in\Omega\times\mathbb{C}^n$, the equation $F(z)=c$ has exactly one solution. His paper explicitly left open which alternative holds for any concrete component, and raised the possibility that JC fails for some pair $(V_n^k,\Omega)$ - in which case, by the dichotomy, counterexamples would be dense in that component and 'nearly impossible to miss'.

One month later the conjecture fell in dimension three. On 19 July 2026 Alpöge announced an explicit Keller counterexample $F:\mathbb{C}^3\to\mathbb{C}^3$ (a polynomial map of degree seven with $\det DF\equiv-2$ that identifies three distinct points). On 20 July Gallagher gave a uniform construction of counterexamples $F_m:\mathbb{C}^3\to\mathbb{C}^3$ with $\det DF_m\equiv1$ and generic fiber degree - the number of preimages of a general point - exactly $m$, for every $m\ge3$. Expositions by Tao and by Speyer followed within days. Speyer isolated the mechanism, the tangent sweep: sweeping out the tangent lines of a plane curve is unavoidably many-to-one (by classical projective duality a general point of the dual plane lies on several tangent lines), and judicious monomial conjugations cancel the Jacobian factor picked up by the sweep at the cost of pushing all ramification to infinity; the resulting Keller maps are everywhere locally invertible (in algebraic geometry language, étale, here just meaning the Jacobian never vanishes) but not proper - some sequences leave every compact set while their images stay bounded - and they fail injectivity only through this escape to infinity. Gao gave a self-contained account, generalized the construction to direction fields on hypersurfaces, and produced explicit counterexamples in every dimension $n>2$ (of geometric degrees 4, 5, 10, 6, and 12 in dimensions three through five), with all identities verified in exact rational arithmetic and fiber structures computed by Gröbner bases (standard terminating algorithms for polynomial systems). Consequently JC is now known to be false in every dimension $n\ge3$ (composing a three-dimensional counterexample with the identity gives one in each higher dimension); the case $n=1$ is trivial; and dimension two remains open - Meng and Yang record that of the Jacobian family only the two-dimensional statement $JC_2$ is still unsettled.

In parallel, Jelonek, apparently independently of Xavier, proved that the set $A(n,d)$ of polynomial automorphisms of degree at most $d$ with Jacobian determinant one is Zariski closed, recovered the same component dichotomy for the variety $X(n,d)$ (his notation for $V_n^d$), and added two structural facts: every irreducible component of $X(n,d)$ has dimension at least $n^2-1$ (each component is stable under right composition with the $n^2-1$-dimensional group of determinant-one linear maps), and if $X(n,d)$ is irreducible for some $n\ge3$ and $d\ge6$, then a generic element of it is a counterexample to JC. Drużkowski and Rusek had already suggested in 1983, for a special subvariety of maps in Drużkowski form (maps $x+(Ax)^{*d}$ built from powers of linear forms), that such varieties might be reducible; the general irreducibility question remains open.

For the component-level problem, the dichotomy turns each explicit counterexample into a component-level statement: the irreducible component of $V_n^k$ containing any known counterexample consists, generically, of counterexamples, so the second alternative genuinely occurs. After a harmless linear rescaling (composing with a diagonal linear map of determinant $-1/2$) one obtains maps with $\det\equiv1$ and degree at most seven in every dimension $n\ge3$, so $V_n^k$ contains counterexample components for all $k\ge7$. What is not known is the complete determination: in dimension two, where by the dichotomy the determination for all components of all $V_2^k$ settles the still-open $JC_2$ - a proof of $JC_2$ would show every planar component to consist of automorphisms, while a planar counterexample would fix, via the dichotomy, only the components containing it; and in dimensions $n\ge3$, where no component's alternative has been pinned down beyond the components containing explicit counterexamples - even the component containing the identity (or the determinant-one linear maps) has not been shown to consist of automorphisms, since the dichotomy permits a component to contain some automorphisms and still be generically non-invertible. The graded (weight-equivariant) viewpoint of Shaska organizes the known counterexamples - the announced example is equivariant for the weight grading $\mathrm{wt}(x,y,z)=(1,-1,-2)$ - and shows that in dimension two every graded Keller map is an automorphism, so a planar counterexample, if one exists, must come from a genuinely new mechanism.

Problem Statement

Determine, for every $n\ge2$, every $k\ge1$, and every irreducible component $\Omega$ of the variety $V_n^k=\{F:\mathbb{C}^n\to\mathbb{C}^n$ polynomial with $\deg F\le k$ and $\det DF\equiv1\}$, which alternative of the Xavier-Jelonek dichotomy holds for $\Omega$: either $\Omega\subset\mathrm{Aut}(\mathbb{C}^n)$ (every map in the component is a polynomial automorphism), or the general $F\in\Omega$ is a Keller counterexample (a non-automorphism with constant nonzero Jacobian determinant).

The determination currently splits into two open sectors.

1. Dimension two: decide, for every component of every $V_2^k$, which alternative holds. By the dichotomy, a proof of the two-dimensional Jacobian conjecture $JC_2$ - that every planar Keller map is invertible, the last open case of the Jacobian conjecture - would show every such component to consist of automorphisms, while a single planar Keller counterexample would show via the dichotomy that the components containing it are generically counterexample components, but would leave the remaining components to be determined.

2. Dimensions $n\ge3$: classify the components of $V_n^k$ for each $(n,k)$ - which components are automorphism components and which are generically counterexample components; in particular, decide for each concrete $(n,d)$ whether $X(n,d)$ (Jelonek's notation for $V_n^d$) is irreducible or reducible (Jelonek proved that an irreducible $X(n,d)$ with $n\ge3$ and $d\ge6$ has generic element a counterexample), and determine the status of specific components such as the component containing the identity or the determinant-one linear maps.

The known boundary facts constrain any determination: the dichotomy theorem itself; the existence of counterexample components in every dimension $n\ge3$ for all $k\ge7$ (Alpöge, Gallagher, Gao); invertibility of all maps for $n=1$ and for $k\le2$ (Wang's quadratic case); and $\dim\Omega\ge n^2-1$ for every component (Jelonek).

References

1. F. Xavier, An organizing principle in the study of the Jacobian Conjecture, arXiv:2606.22041 (2026). DOI: 10.48550/arxiv.2606.22041, https://arxiv.org/abs/2606.22041
2. Z. Jelonek, On mappings with Jacobian one, arXiv:2607.20597 (2026). DOI: 10.48550/arxiv.2607.20597, https://arxiv.org/abs/2607.20597
3. S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222 (2026). DOI: 10.48550/arxiv.2608.00222, https://arxiv.org/abs/2608.00222
4. A. Gallagher, An infinite family of counterexamples to the Jacobian Conjecture in dimension three: every generic fiber degree $n\ge3$ occurs, preprint (2026). DOI: 10.5281/zenodo.21479195, https://zenodo.org/records/21479195
5. L. Alpöge, A counterexample to the Jacobian conjecture in dimension three, announcement, 19 July 2026, https://x.com/__alpoge__/status/2079028340955197566
6. T. Tao, A digestion of the Jacobian conjecture counterexample, blog post, 21 July 2026, https://terrytao.wordpress.com/2026-07-21/a-digestion-of-the-jacobian-conjecture-counterexample/
7. D. E. Speyer, The geometry and structure of Gallagher's counterexamples to the Jacobian conjecture, note, 23 July 2026, https://sbseminar.wordpress.com/wp-content/uploads/2026/07/jacobiantangentsweep.pdf
8. T. Shaska, Graded Keller maps and the Jacobian Conjecture, arXiv:2607.20210 (2026). DOI: 10.48550/arxiv.2607.20210, https://arxiv.org/abs/2607.20210
9. K. Kistner and T. Shaska, Orbits and Fields of Definition for Graded Keller Maps, arXiv:2608.02863 (2026). DOI: 10.48550/arxiv.2608.02863, https://arxiv.org/abs/2608.02863
10. G. Meng and L. Yang, A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures, arXiv:2607.22198 (2026). DOI: 10.48550/arxiv.2607.22198, https://arxiv.org/abs/2607.22198
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12. H. Bass, E. H. Connell and D. Wright, The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bulletin of the American Mathematical Society (N.S.) 7 (1982) 287-330. DOI: 10.1090/S0273-0979-1982-15032-7
13. S. S.-S. Wang, A Jacobian criterion for separability, Journal of Algebra 65 (1980) 453-494. DOI: 10.1016/0021-8693(80)90233-1
14. L. M. Drużkowski, An effective approach to Keller's Jacobian Conjecture, Mathematische Annalen 264 (1983) 303-313. DOI: 10.1007/BF01459126
15. K. Rusek, A geometric approach to Keller's Jacobian Conjecture, Mathematische Annalen 264 (1983) 315-320. DOI: 10.1007/BF01459127