Open problem, Algebra, Analysis, posed 1939

Jacobian conjecture

Partially solvedSmale #16

Let $k$ be a field of characteristic $0$ (equivalently $k = \mathbb{C}$ ) and let $F = (f_1, \dots, f_n) : k^n \to k^n$ be a polynomial mapping whose Jacobian determinant $\det\!\left(\frac{\partial f_i}{\partial x_j}\right)$ is a nonzero constant in $k^\times$ . Must $F$ have a polynomial inverse $G : k^n \to k^n$ ?

Research frontier as of 2026

In July 2026, Levent Alpöge disproved the general Jacobian conjecture in all dimensions $n \ge 3$ by exhibiting the explicit polynomial map $F(x,y,z) = ((1+xy)^3 z + y^2(1+xy)(4+3xy),\, y + 3x(1+xy)^2 z + 3xy^2(4+3xy),\, 2x - 3x^2 y - x^3 z)$ , which has constant Jacobian determinant $-2$ everywhere on $\mathbb{C}^3$ but is generically $3$ -to- $1$ . Consequently, only the two-dimensional planar Jacobian conjecture ( $n = 2$ ) remains open as of 2026.

Best known results

Alpöge (2026): the Jacobian conjecture is false in every dimension $n \ge 3$ , with an explicit counterexample in $n = 3$ of degree $5$ .
In dimension $n = 2$ , the conjecture holds whenever the field extension $\mathbb{C}(x,y)/\mathbb{C}(f,g)$ is Galois (Campbell, 1973), when the degree is at most $104$ (Moh, 1983; Nguyen, 2025), or when $\gcd(\deg f, \deg g)$ is a prime or at most $16$ (Abhyankar, Appelgate–Onishi, Nagata).

Tools and where they stop

Tool	Achieved	Where it stops
Newton polygons and approximate roots at infinity (Abhyankar–Moh–Nagata)	Rules out planar counterexamples up to degree $104$ and constrains the ratio $\deg f / \deg g$ of any hypothetical planar counterexample	Combinatorial explosion of singular branches at infinity prevents covering arbitrary composite degree pairs in $n = 2$
Degree reduction and nilpotent Jacobian matrices (Bass–Connell–Wright)	Reduces the general problem to cubic-homogeneous maps $x + H(x)$ at the cost of increasing the dimension $n$	Because stabilization increases the dimension $n$ and the conjecture fails for $n \ge 3$ , high-dimensional degree reduction cannot settle $n = 2$

Open questions

Does the planar Jacobian conjecture ( $n = 2$ ) hold for all polynomial maps $F : \mathbb{C}^2 \to \mathbb{C}^2$ with constant nonzero Jacobian determinant?
Does the first Dixmier conjecture hold: is every algebra endomorphism of the first Weyl algebra $A_1(\mathbb{C})$ an automorphism?

References

Ott-Heinrich Keller (1939). Ganze Cremona-Transformationen · DOI:10.1007/BF01704923
Hyman Bass, Edwin H. Connell, David Wright (1982). The Jacobian conjecture: reduction of degree and formal expansion of the inverse · DOI:10.1090/S0273-0979-1982-15032-7
Arno van den Essen (2000). Polynomial Automorphisms and the Jacobian Conjecture