The two-dimensional Jacobian conjecture and the four-dimensional Hessian conjecture over fields of characteristic zero

Keller's Jacobian conjecture $\mathrm{JC}_n$ and Meng's Hessian conjecture $\mathrm{HC}_n$ form two dimension-indexed families of problems about polynomial maps and polynomials with constant nonzero Jacobian or Hessian determinant. After Alpöge's July 2026 refutation of $\mathrm{JC}_3$ and Meng–Yang's resulting refutation of $\mathrm{HC}_5$ (via doubling and a one-variable Schur descent), exactly two statements of the two families over fields of characteristic zero remain undecided: the two-dimensional Jacobian conjecture $\mathrm{JC}_2$, open since 1939, and the four-dimensional Hessian conjecture $\mathrm{HC}_4$. They are linked by the implication $\mathrm{HC}_4 \Rightarrow \mathrm{JC}_2$, so a proof of $\mathrm{HC}_4$ would prove $\mathrm{JC}_2$ and a counterexample to $\mathrm{JC}_2$ would refute $\mathrm{HC}_4$; no converse implication is known. The problem is to decide both surviving statements — prove or refute $\mathrm{HC}_4$ (in unrestricted degree) and prove or refute $\mathrm{JC}_2$ — thereby completing the truth classification of the two families. The quartic case of $\mathrm{HC}_4$ was proved by Ni (2026), leaving the general statement open.

Background

A polynomial map $F = (F_1, \dots, F_n) : k^n \to k^n$ over a field $k$ is called a Keller map if its Jacobian determinant $\operatorname{Jac} F = \det(\partial F_i / \partial x_j)$ is a nonzero constant. Keller's 1939 Jacobian conjecture $\mathrm{JC}_n$ asserts that every Keller map is invertible with a polynomial inverse; the inverse function theorem gives only a local analytic inverse, and the conjecture was open in every dimension $n \ge 2$ from 1939 until 2026. Classical reductions (Bass–Connell–Wright's reduction of degree and formal inverse expansion; de Bondt–van den Essen's reduction to symmetric maps) shaped the problem but did not settle it.

The Hessian conjecture $\mathrm{HC}_n$, introduced by Meng (2006), is a companion family. For a polynomial $\varphi \in k[x_1, \dots, x_n]$ whose Hessian determinant $\det \operatorname{Hess} \varphi$, with $\operatorname{Hess} \varphi = (\partial^2 \varphi / \partial x_i \partial x_j)$ the matrix of second partial derivatives, is a nonzero constant, the gradient map $\nabla \varphi$ has a formal inverse near any base point, and the formal Legendre transform $\varphi^L$ is defined classically by $\varphi^L(p) = \langle p, x(p) \rangle - \varphi(x(p))$ where $x(p)$ is that formal inverse; in general $\varphi^L$ is a formal power series. $\mathrm{HC}_n$ asserts that $\varphi^L$ is always a polynomial, which is equivalent to $\nabla \varphi$ having a polynomial inverse. The two families are bridged by two elementary implications due to Meng: $\mathrm{JC}_n \Rightarrow \mathrm{HC}_n$ (a polynomial inverse of $\nabla \varphi$ makes $\varphi^L$ a polynomial) and the doubling implication $\mathrm{HC}_{2n} \Rightarrow \mathrm{JC}_n$ (applied to the doubled potential $\varphi(x, y) = \langle y, F(x) \rangle$, whose Hessian determinant is $(-1)^n (\operatorname{Jac} F)^2$). Falsehood propagates upward in both families by trivial stabilization (conjugating a counterexample, or appending identity variables to it, yields one in every higher dimension).

Two 2026 developments collapsed the landscape. On 19 July 2026 Levent Alpöge announced an explicit Keller map $F : \mathbb{C}^3 \to \mathbb{C}^3$ (constructed by the AI system Fable after Akhil Mathew suggested the problem) with $\operatorname{Jac} F \equiv -2$ and three distinct rational points sharing an image; both facts are finite exact computations, independently reproduced, and the community quickly built a geometric model of the example via the tangent-line geometry of $\mathbb{P}^1 \times \operatorname{Sym}^2(\mathbb{P}^1) \to \operatorname{Sym}^3(\mathbb{P}^1)$ (where $\operatorname{Sym}^n(\mathbb{P}^1)$ denotes the quotient variety parametrizing unordered $n$-tuples of points on the projective line $\mathbb{P}^1$) and generalizations to every dimension above two. Thus $\mathrm{JC}_n$ is false for all $n \ge 3$ in characteristic zero, while $\mathrm{JC}_2$ is untouched, since no reduction from dimension 3 down to dimension 2 is known. Meng and Yang then applied the doubling construction to Alpöge's map followed by a one-variable Schur descent (a partial Legendre transform eliminating a pivot variable), obtaining an explicit five-variable integer polynomial of total degree 14 with constant Hessian determinant 128 and non-injective gradient, refuting $\mathrm{HC}_5$; stabilization refutes $\mathrm{HC}_n$ for all $n \ge 5$. Combined with de Bondt's theorem that $\mathrm{HC}_n$ holds for $n \le 3$, the truth table of both families over characteristic-zero fields is decided everywhere except two entries: $\mathrm{JC}_2$ (open) and $\mathrm{HC}_4$ (open), linked by the single bridge $\mathrm{HC}_4 \Rightarrow \mathrm{JC}_2$. Meng and Yang observe that their descent mechanism self-terminates and cannot reach four variables, so $\mathrm{HC}_4$ appears to require genuinely new ideas; in the other direction, the class-group analysis of the new geometric constructions shows the tangent-sweep mechanism exists only in dimension three and above.

Problem Statement

Over a field $k$ of characteristic zero, decide the two remaining unsettled statements of the Jacobian and Hessian conjecture families:

(i) $\mathrm{HC}_4$ (four-dimensional Hessian conjecture): prove or refute that every polynomial $\varphi \in k[x_1, x_2, x_3, x_4]$ with constant nonzero Hessian determinant has a polynomial formal Legendre transform (equivalently, has a gradient map $\nabla \varphi : k^4 \to k^4$ with a polynomial inverse). The quartic case is known to hold; the statement must be settled for polynomials of arbitrary degree.

(ii) $\mathrm{JC}_2$ (two-dimensional Jacobian conjecture, Keller 1939): prove or refute that every polynomial map $F : k^2 \to k^2$ whose Jacobian determinant is a nonzero constant has a polynomial inverse.

The two statements are linked by the known doubling implication $\mathrm{HC}_4 \Rightarrow \mathrm{JC}_2$: a proof of $\mathrm{HC}_4$ would prove $\mathrm{JC}_2$, and a counterexample to $\mathrm{JC}_2$ would refute $\mathrm{HC}_4$; no implication in the other directions is known. The problem is completely resolved when both statements are decided (each proved or refuted); a single explicit counterexample to $\mathrm{JC}_2$ would resolve both at once, as would a single proof of $\mathrm{HC}_4$. Refuting $\mathrm{HC}_4$ alone, or proving $\mathrm{JC}_2$ alone, decides one statement and leaves the other as the surviving remnant. Characteristic zero is essential to the statement: the corresponding separable Jacobian conjecture in characteristic 2 is already known to fail in dimension two.

References

1. G. Meng, 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
2. O.-H. Keller, "Ganze Cremona-Transformationen", Monatshefte für Mathematik und Physik 47 (1939), 299–306, DOI 10.1007/BF01695502, https://doi.org/10.1007/BF01695502
3. H. Bass, E. H. Connell, D. Wright, "The Jacobian conjecture: Reduction of degree and formal expansion of the inverse", Bulletin of the American Mathematical Society 7 (1982), 287–330, DOI 10.1090/S0273-0979-1982-15032-7, https://doi.org/10.1090/S0273-0979-1982-15032-7
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6. M. de Bondt, "Polynomials with constant Hessian determinants in dimension three", Journal of Pure and Applied Algebra 219 (2015), 3743–3754, arXiv:1203.6605, DOI 10.1016/j.jpaa.2014.12.020, https://arxiv.org/abs/1203.6605
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8. A. Gallagher, "An infinite family of counterexamples to the Jacobian Conjecture in dimension three: every generic fiber degree $n \ge 3$ occurs", Zenodo preprint, 20 July 2026, DOI 10.5281/zenodo.21479195, https://zenodo.org/records/21479195 (author's companion explainer: https://jacobianfun.org/jacobian-explained)
9. S. Gao, "Counterexamples to the Jacobian conjecture in dimensions greater than two", arXiv:2608.00222 (2026), https://arxiv.org/abs/2608.00222
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