The two-variable Jacobian conjecture asks whether two complex polynomials f,g∈C[x,y]f,g\in\mathbb{C}[x,y] whose Jacobian determinant is identically 11 — a Jacobian pair, ff being a Jacobian mate — must generate the full polynomial ring, C[f,g]=C[x,y]\mathbb{C}[f,g]=\mathbb{C}[x,y]; equivalently, whether every polynomial self-map of the complex plane with constant nonzero Jacobian is invertible with a polynomial inverse. The problem originates in Keller's 1939 work. In July 2026 the conjecture was refuted in every dimension three and higher (Alpöge's explicit three-variable counterexample and subsequent generalizations), which makes this planar case — apart from the four-variable Hessian conjecture, which implies it — the sole surviving open statement of Keller's problem. Makar-Limanov's 2025 Israel Journal of Mathematics note states the question as not known.

Background

The Jacobian conjecture is a 1939 problem of Ott-Heinrich Keller about polynomial maps of the affine plane. Write C[x,y]\mathbb{C}[x,y] for the ring of complex polynomials in two commuting variables. For f,g∈C[x,y]f,g\in\mathbb{C}[x,y], the Jacobian J(f,g)J(f,g) is the determinant of the matrix of partial derivatives, ∂f∂x∂g∂y−∂f∂y∂g∂x\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}. If the map (f,g):C2→C2(f,g):\mathbb{C}^2\to\mathbb{C}^2 has a polynomial inverse (a polynomial automorphism), the chain rule forces J(f,g)J(f,g) to be a nonzero constant, and rescaling gg makes it identically 11. Keller's problem asks the converse: must such a pair be invertible? In one variable the answer is immediate, because a univariate polynomial with constant derivative is linear; in two variables the question has been open since 1939.

The problem has a purely algebraic formulation. The subalgebra C[f,g]\mathbb{C}[f,g] generated by ff and gg equals all of C[x,y]\mathbb{C}[x,y] exactly when (f,g)(f,g) is a polynomial automorphism, so the conjecture equivalently asks: if J(f,g)≡1J(f,g)\equiv 1, must C[f,g]=C[x,y]\mathbb{C}[f,g]=\mathbb{C}[x,y]? Following Makar-Limanov, ff is called a Jacobian mate if such a gg exists, and (f,g)(f,g) is a Jacobian pair. This is precisely the formulation of a 2025 note of Makar-Limanov in the Israel Journal of Mathematics, which states it as an open question and identifies it with the Jacobian conjecture.

The conjecture became a central problem of affine algebraic geometry through the survey of Bass, Connell and Wright (1982), which standardized the modern formulation and the main reductions, for example to maps of low degree. It should not be confused with the strong real Jacobian conjecture — the statement that a real polynomial map R2→R2\mathbb{R}^2\to\mathbb{R}^2 whose Jacobian merely never vanishes (without being constant) is injective — which Pinchuk refuted in 1994; the complex constant-Jacobian question studied here is untouched by that counterexample.

A dramatic recent development frames the current status. On 19 July 2026, Levent Alpöge announced an explicit polynomial map C3→C3\mathbb{C}^3\to\mathbb{C}^3 with constant Jacobian −2-2 that is not injective, refuting the conjecture in dimension three; Alexis Gallagher produced an infinite family, David Speyer and Terence Tao posted expositions (Tao stating explicitly that the conjecture remains open in two dimensions), and Shuhong Gao generalized the construction to every dimension greater than two. The refuting mechanism — essentially a parametrized sweep of the tangent lines of a plane curve — is intrinsically at least three-dimensional, and Meng and Yang showed that among the Jacobian and Hessian conjecture families exactly two statements survive: the two-variable Jacobian conjecture considered here, and the four-variable Hessian conjecture — the analogous statement that a polynomial map Cn→Cn\mathbb{C}^n\to\mathbb{C}^n whose Hessian determinant (the determinant of its matrix of second partial derivatives) is a nonzero constant must be invertible — which implies it. The two-variable case is therefore now the last open instance of Keller's problem, and the mate literature continues to address it directly: Makar-Limanov (2025) proved that for fixed ff the subalgebra C[f,g]\mathbb{C}[f,g] is determined and gg is recoverable up to a summand that is a polynomial in ff, and Makar-Limanov and Trakhtenberg (2026) derived further restrictions on the mate of a hypothetical counterexample while explicitly stating that the conjecture itself is not known.

Problem Statement

Let f,g∈C[x,y]f,g\in\mathbb{C}[x,y] be complex polynomials in two variables whose Jacobian determinant is identically 11: J(f,g)  =  ∂f∂x∂g∂y−∂f∂y∂g∂x  ≡  1.J(f,g)\;=\;\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\;\equiv\;1. Must the subalgebra generated by ff and gg be the entire polynomial ring, C[f,g]=C[x,y]\mathbb{C}[f,g]=\mathbb{C}[x,y]? Equivalently: must the map (f,g):C2→C2(f,g):\mathbb{C}^2\to\mathbb{C}^2 be a polynomial automorphism (invertible with a polynomial inverse)? This is the two-variable Jacobian conjecture, the last open statement of Keller's 1939 problem after the July 2026 refutation of the conjecture in every dimension at least three.

References

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