The Jacobian Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\in\mathbb N$ and let $F=(F_1,\ldots,F_n):\mathbb C^n\to\mathbb C^n$ be a polynomial map, meaning $F_i\in\mathbb C[x_1,\ldots,x_n]$ for every $i$. Its Jacobian matrix and determinant are \[ J_F(x)=\left(\frac{\partial F_i}{\partial x_j}(x)\right)_{1\le i,j\le n}, \qquad \det J_F(x). \] A polynomial automorphism means a polynomial map with a polynomial two-sided inverse: some $G:\mathbb C^n\to\mathbb C^n$ with $G\circ F=F\circ G=\operatorname{id}_{\mathbb C^n}$.

Statement recorded in the supplied source.

The historical Jacobian conjecture asserts that, for every $n\in\mathbb N$ and every such polynomial map $F$, if there is a number $c\in\mathbb C\setminus\{0\}$ such that \[ \det J_F(x)=c\qquad\text{for every }x\in\mathbb C^n, \] then $F$ is a polynomial automorphism. Is this implication valid in every dimension? The inverse is required to be polynomial, rather than merely a local holomorphic inverse. [S2]

Short English statement

Does every polynomial map of complex affine space whose Jacobian determinant is a nonzero constant have a polynomial inverse? The assertion quantifies over all positive dimensions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 54 (SMA-016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Shuhong Gao, Counterexamples to the Jacobian Conjecture in Dimensions Greater Than Two, arXiv:2608.00222v1 (2026), Section 1, formulation of the historical conjecture, pages 1--2. https://arxiv.org/pdf/2608.00222.