Jacobian Conjecture Overview

Updated 21 November 2025

Jacobian Conjecture is an open problem in affine algebraic geometry examining when polynomial maps with a constant, nonzero Jacobian determinant are automorphisms.
Key reductions like the Bass–Connell–Wright and Drużkowski methods simplify the problem and connect it to related fields such as commutative algebra and topology.
Research employs algebraic, combinatorial, and topological techniques, revealing deep ties to conjectures like the Dixmier Conjecture and impacting broader mathematical theories.

The Jacobian Conjecture is a central unsolved problem in affine algebraic geometry concerning the invertibility of polynomial maps with constant Jacobian determinant. Formulated in the 1930s and pursued extensively since, it sits at the intersection of algebraic geometry, commutative algebra, topology, and mathematical physics, and is connected to several other major conjectures in mathematics.

1. Formulation and Core Statement

Given a field $k$ of characteristic zero, the Jacobian Conjecture considers a polynomial endomorphism $F: k^n \to k^n$ given by

$F(x) = (f_1(x), \ldots, f_n(x)),$

where $f_i \in k[x_1, \ldots, x_n]$ . The associated Jacobian matrix is $J(F)(x) = (\partial f_i / \partial x_j)$ , and its determinant $J(F)(x) \in k[x_1, ..., x_n]$ .

Jacobian Conjecture:

If $\det J(F)(x) \in k^*$ (i.e., the Jacobian is a nonzero constant), then $F$ is an automorphism of the polynomial algebra; i.e., $F$ is invertible and the inverse is also a polynomial map.

For $n=2$ , the conjecture specializes to automorphisms of $k[x,y]$ , and for $n>2$ it remains open and highly nontrivial (Miyanishi, 2015, Moskowicz, 2015).

2. Historical Developments and Reductions

The conjecture was first formulated explicitly by O. H. Keller in 1939 for $k = \mathbb{C}$ and $n = 2$ . Following an incorrect proof, the problem has resisted resolution for over 80 years (Miyanishi, 2015).

Major structural reductions include:

Bass–Connell–Wright reduction: It suffices to prove the conjecture for polynomial maps of the form $F = X + H$ , with $H$ homogeneous of degree 3 and $JH$ nilpotent (Mendson, 2018, Miyanishi, 2015).
Drużkowski reduction: It further suffices to treat $F$ where each coordinate of $H$ is the cube of a linear form, and associated nilpotency constraints on the Jacobian (Adamus et al., 2015, Liu, 2020).
In dimension $2$, Wang showed invertibility when all coordinate degrees are at most 2; see also Moh’s result for degrees up to 100, and Magnus–Nakai–Baba for the prime-degree case (Miyanishi, 2015).

3. Structural, Algebraic, and Geometric Features

The conjecture is intimately related to properties of affine spaces, unramified (étale) morphisms, and the structure of algebraic varieties:

Étale morphisms: A polynomial map with constant nonzero Jacobian is an étale morphism; the conjecture is equivalent to every such endomorphism of $\mathbb{A}^n$ being an automorphism (Miyanishi, 2015, Adjamagbo, 2012).
Ring-theoretic conditions: In two variables, the problem can be equivalently reformulated in terms of normality, flatness, or separability properties of intermediate subalgebras such as $k[f(x), f(y)][x+y]$ (Moskowicz, 2015).
Topological techniques: Approaches using compactifications, the paper of the boundary divisor and its dual graph, and dynamical (Hamiltonian vector field) arguments in dimension 2 (notably over $\mathbb{R}$ ) reveal deep connections to affine and projective surface theory (Zhang, 2020, Miyanishi, 2015).

4. Methodological Approaches and Special Cases

Multiple methodologies have established the conjecture in restricted settings or provided significant insight:

Low-degree and parity cases: In dimension two, invertibility follows if one coordinate has degree $\leq 2$ or all monomial degrees exhibit parity constraints (Moskowicz, 2015).
Involutive symmetry: If a map commutes with a coordinate-exchange involution (or can be conjugated to such a form), invertibility holds (Moskowicz, 2014, Moskowicz, 2015).
Nilpotency and algorithmic inversion: Cubic polynomial endomorphisms with strongly nilpotent Jacobian matrices can be explicitly inverted with bounded degree, suggesting a strong connection between Jacobian nilpotency and automorphicity (Adamus et al., 2015).
Tree-theoretic and combinatorial models: Formal inversion formulas and shuffle identities for tree-structured expansions allow a combinatorial reframing, closely tied to the nilpotency condition and leading to approximate results (Bisi et al., 2023).
Partial differential equation viewpoint: The Jacobian condition translates into a highly structured nonlinear system, allowing explicit construction of infinite families of automorphisms and a full solution for multiply-parameterized variants (Yang, 2021).
Arithmetic methods: There is an equivalence between the global conjecture and the existence of unimodular primes—those for which every Keller map over $\mathbb{Z}_p$ descends non-trivially mod $p$ (Mendson, 2018).

5. Connections to Broader Conjectures and Algebraic Structures

The Jacobian Conjecture is equivalent, in a stable sense, to other major problems:

Dixmier Conjecture: The assertion that every endomorphism of the $n$ th Weyl algebra is an automorphism is equivalent to the Jacobian Conjecture, up to stabilization (Bavula, 2021, Miyanishi, 2015).
Poisson Conjecture: For polynomial Poisson algebras, the automorphism property for Poisson endomorphisms implies the Jacobian Conjecture (Bavula, 2021).
Holonomic module criteria: The invertibility of Keller maps is detected by the 1-generation (cyclicity) of certain twisted modules over the Weyl algebra; more generally, these modules are always holonomic and cyclic, but the conjecture asserts simplicity (Bavula, 2021).

6. Counterexamples to Generalizations and Partial Negative Results

While the classical Jacobian Conjecture remains unresolved, its geometric extensions have known counterexamples:

Bass Generalized Jacobian Conjecture: There exist smooth, unirational, affine surfaces (e.g., certain complements in blowups of $\mathbb{P}^2$ ) admitting unramified morphisms of degree $>1$ to affine space, thus negating GJC in dimension $\geq2$ . However, these surfaces are not isomorphic to affine space (Adjamagbo, 2012).
Real case and non-finite endomorphisms: Over $\mathbb{R}$ , Pinchuk constructed polynomial r-maps of the plane with nonvanishing Jacobian which are not surjective (Miyanishi, 2015).

7. Open Problems and Current Directions

Despite major advances in reductions and special case proofs, the full Jacobian Conjecture in dimension $n \geq 2$ remains open. Major lines of research include:

Further reduction to the quadratic case via algebraic or QFT-inspired techniques (Goursac et al., 2014).
Refinement of combinatorial and tree-based criteria and their connections to nilpotency (Bisi et al., 2023, Adamus et al., 2015).
Deepening understanding of field extension degrees, minimal polynomial models, and their role in function field formulations (Moskowicz, 2016).
Investigation of arithmetic analogues and their implications for the conjecture over finite and $p$ -adic fields (Mendson, 2018), construction of probabilistic and asymptotic analogues.
Exploration of the geometry and topology at infinity, including the paper of fundamental groups, boundary divisors, and affine pseudo-coverings (Miyanishi, 2015).
Progress on related conjectures in noncommutative algebra and their interplay with the Jacobian Conjecture (Bavula, 2021).

The Jacobian Conjecture continues to serve as a proving ground for advanced tools from algebraic geometry, ring theory, topology, PDE theory, and quantum field theory, and its resolution is expected to have major repercussions across these domains.