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Jacobian conjecture

The Jacobian conjecture is a longstanding unsolved problem in algebraic geometry and commutative algebra, asserting that a polynomial map F: \mathbb{C}^n \to \mathbb{C}^n is a polynomial automorphism (i.e., bijective with a polynomial inverse) if and only if the determinant of its Jacobian matrix is a nonzero constant.^[1]^[2] Formulated by Otto H. Keller in 1939, the conjecture originally concerned polynomial maps over the integers but was soon generalized to fields of characteristic zero, such as the complex numbers.^[2] It holds trivially in one dimension (n=1), as a polynomial with constant nonzero derivative is affine, but remains open for n \geq 2.^[1] Equivalent formulations include the conditions that such a map is injective or proper, linking the conjecture to broader questions in topology and dynamics.^[1] Significant progress has been made through reductions: the conjecture is proven for maps of degree at most 2, and it has been reduced to the case of degree 3 maps of the form F = X - H where the Jacobian of H is nilpotent.^[2] In dimension 2, it holds for degrees up to 108 (as of 2022), but no general proof exists despite extensive efforts involving combinatorial, analytic, and perturbative methods.^[2]^[3] As of 2025, the conjecture continues to inspire research, with recent work exploring arithmetic interpretations and formal power series expansions, yet it stands unresolved.^[4]^[5]

Mathematical Background

Jacobian matrix and determinant

The Jacobian matrix of a differentiable map F: \mathbb{R}^n \to \mathbb{R}^n at a point x = (x_1, \dots, x_n) is the n \times n matrix whose (i,j)-th entry is the partial derivative \frac{\partial F_i}{\partial x_j}(x), where F = (F_1, \dots, F_n). This matrix, often denoted J_F(x), provides the best linear approximation to the change in F near x, capturing the local behavior of the map through its first-order partial derivatives. The Jacobian determinant, \det(J_F(x)), quantifies the local scaling effect of the map on volumes. Specifically, it measures how the map distorts infinitesimal volume elements: a small parallelepiped of volume dV in the domain is mapped to one of volume |\det(J_F(x))| \, dV in the codomain, with the sign of the determinant indicating orientation preservation or reversal.^[6] For instance, if |\det(J_F(x))| > 1, volumes expand locally, while |\det(J_F(x))| < 1 indicates contraction. A key property arises from the inverse function theorem: if F is continuously differentiable near x and \det(J_F(x)) \neq 0, then F is locally invertible at x, with a continuously differentiable inverse in some neighborhood. The inverse map's Jacobian is then the inverse matrix, J_{F^{-1}}(F(x)) = [J_F(x)]^{-1}. Globally, for a C^1 bijection F: \mathbb{R}^n \to \mathbb{R}^n that is proper (preimages of compact sets are compact) and satisfies \det(J_F(x)) \neq 0 everywhere, F is a diffeomorphism, ensuring a smooth global inverse. Consider simple examples to illustrate. For a linear transformation F(x) = Ax where A is an n \times n matrix, the Jacobian matrix is constantly A, and \det(J_F) = \det(A); local (and global) invertibility holds if and only if \det(A) \neq 0. In one dimension, for a scalar function f: \mathbb{R} \to \mathbb{R}, the Jacobian "matrix" reduces to the derivative f'(x), whose "determinant" is f'(x) itself, and the inverse function theorem states that f is locally invertible at x if f'(x) \neq 0, with volume scaling given by |f'(x)|. These concepts form essential tools for analyzing the local and global invertibility of more general maps, such as polynomial maps on affine space.^[6]

Polynomial maps on affine space

The affine n-space over the complex numbers, denoted \mathbb{A}^n or \mathbb{C}^n, consists of all ordered n-tuples of complex numbers, equipped with the Zariski topology and serving as the ambient space for polynomial mappings.^[1] A polynomial map F: \mathbb{C}^n \to \mathbb{C}^n is defined as an n-tuple F = (F_1, \dots, F_n), where each coordinate function F_i belongs to the polynomial ring \mathbb{C}[x_1, \dots, x_n], meaning each F_i is a finite linear combination of monomials in the variables x_1, \dots, x_n with coefficients in \mathbb{C}.^[1] Such maps are also called polynomial endomorphisms of \mathbb{A}^n, and the degree of F, denoted \deg F, is the maximum of the degrees of its coordinate polynomials \deg F_i.^[7] The invertible polynomial maps, or polynomial automorphisms, correspond precisely to the automorphisms of the polynomial ring \mathbb{C}[x_1, \dots, x_n], which are ring isomorphisms from the ring to itself.^[8] These automorphisms induce bijective polynomial maps on \mathbb{A}^n with polynomial inverses, and by the Jung--van der Kulk theorem (for n=2) and more general structure theorems for higher n, they form the automorphism group \mathrm{Aut}(\mathbb{A}^n).^[9] The group operation corresponds to composition of maps, preserving the polynomial nature.^[8] A key property of polynomial maps is that the composition of two such maps F: \mathbb{C}^n \to \mathbb{C}^n and G: \mathbb{C}^n \to \mathbb{C}^n is again a polynomial map F \circ G: \mathbb{C}^n \to \mathbb{C}^n, with each coordinate of F \circ G obtained by substituting the polynomials of G into those of F.^[1] Moreover, the degree satisfies \deg(F \circ G) \leq (\deg F)(\deg G), with equality holding in generic cases where leading terms do not cancel.^[7] In the context of the Jacobian conjecture, polynomial maps on \mathbb{A}^n are studied particularly when the Jacobian determinant \det J(F) is a non-zero constant, as this condition ensures local invertibility everywhere and motivates questions about global polynomial invertibility.^[1]

Formulation of the Conjecture

Statement over algebraically closed fields of characteristic zero

The Jacobian conjecture, in its standard formulation over algebraically closed fields of characteristic zero, posits a criterion for the invertibility of polynomial maps based on the behavior of their Jacobian determinants. Specifically, let k be an algebraically closed field of characteristic zero, such as the complex numbers \mathbb{C}, and consider a polynomial map F: k^n \to k^n given by F(X) = (F_1(X), \dots, F_n(X)), where each F_i is a polynomial in the variables X = (x_1, \dots, x_n). The conjecture asserts that if the determinant of the Jacobian matrix J(F) of F is a nonzero constant in k^\times, then F is bijective and admits a polynomial inverse.^[2] The hypothesis requires that \det J(F) belongs to k \setminus \{0\} and is independent of the variables x_1, \dots, x_n, meaning it evaluates to the same nonzero value everywhere in k^n. This condition implies that F is locally invertible at every point, as the Jacobian matrix is nonsingular throughout the domain. The conclusion states that F induces an automorphism of the polynomial ring k[x_1, \dots, x_n], so there exists another polynomial map G: k^n \to k^n such that G \circ F = \mathrm{id} and F \circ G = \mathrm{id} on k^n. This ensures global invertibility with a polynomial right and left inverse.^[2]^[10] The motivation for this conjecture lies in bridging local and global properties of polynomial maps, generalizing the well-known fact from linear algebra that a linear map over such a field is invertible if and only if its determinant is nonzero. In the polynomial setting, a constant nonzero Jacobian determinant guarantees local invertibility by the inverse function theorem, but the conjecture extends this to affirm global bijectivity and the existence of a polynomial inverse, thereby characterizing polynomial automorphisms in terms of their Jacobians. This formulation was first proposed by O. H. Keller in 1939 and has since been a central open problem in algebra.^[2]^[10]

Equivalent reformulations

One equivalent reformulation of the Jacobian conjecture states that a polynomial map F: k^n \to k^n, where k is an algebraically closed field of characteristic zero and \det(J_F) is nowhere vanishing, is injective.^[1] This version is stronger than the original formulation assuming a constant non-zero Jacobian determinant, but the two are logically equivalent under the polynomial setting because injectivity of such a map implies that \det(J_F) must be constant.^[1] Another equivalent reformulation, due to Bass, Connell, and Wright, asserts that if F: k^n \to k^n is a polynomial map with constant non-zero Jacobian determinant, then the polynomial ring k[X_1, \dots, X_n] is a finitely generated module over the subring k[F_1, \dots, F_n].^[11] This module-finiteness perspective highlights the algebraic dependence of the original variables on the image coordinates and aids in reductions to lower degrees.^[12] A further equivalent version involves the global generation of the inverse map. Specifically, if \det(J_F) = c \neq 0 is constant, the conjecture holds if and only if the formal power series inverse of F, constructed via the adjugate matrix of J_F scaled by $1/c, terminates as a polynomial, thereby yielding an explicit polynomial inverse.^[12] This reformulation emphasizes the convergence of the formal expansion to a polynomial under the conjecture's assumptions.^[13]

Partial Results and Known Cases

Trivial and low-dimensional cases

The Jacobian conjecture holds trivially in the one-dimensional case over the complex numbers. For a polynomial map f: \mathbb{C} \to \mathbb{C} such that the Jacobian determinant, which is simply the derivative f'(z), is a nonzero constant, f must be an affine linear function of the form f(z) = az + b with a \neq 0. Such functions are clearly bijective, with the inverse g(z) = (z - b)/a also a polynomial, verifying the conjecture without advanced tools.^[2]^[1] In higher dimensions, the conjecture is immediately true for linear maps. Consider a linear polynomial map F: \mathbb{C}^n \to \mathbb{C}^n given by F(\mathbf{x}) = A\mathbf{x} + \mathbf{b}, where A is an n \times n matrix with constant nonzero determinant \det(A) \neq 0. By standard results in linear algebra, F is an affine isomorphism, hence bijective, and its inverse is also affine linear, thus a polynomial map. This establishes the conjecture for all degree-1 polynomials over \mathbb{C}, as their Jacobians are constant matrices equal to A, and the condition requires \det(A) \neq 0.^[2] An analogous statement holds over the real numbers in one dimension, though with nuances regarding the real Jacobian conjecture. For a polynomial f: \mathbb{R} \to \mathbb{R} with constant nonzero derivative f'(x) = a \neq 0, f is again affine linear f(x) = ax + b. If a > 0, f is strictly increasing and bijective onto \mathbb{R}; if a < 0, it is strictly decreasing and also bijective. The inverse remains a polynomial, satisfying the conjecture trivially, but real versions often emphasize properness or injectivity due to potential non-surjectivity in higher dimensions or for non-constant Jacobians.^[14]

Results for specific degrees and variables

The Jacobian conjecture has been affirmed for polynomial maps of degree 2 in any number of variables. This was proven by S. S. Wang using symmetrization techniques to show that such maps are invertible with polynomial inverses. For maps in two variables, the conjecture holds for degrees less than 100, as established by T. T. Moh using algebraic methods.^[2] Significant reductions simplify the conjecture by showing that its validity for higher degrees implies it for lower degrees. Specifically, the work of H. Bass, E. H. Connell, and D. Wright demonstrates that the conjecture in arbitrary degree is equivalent to the case of maps of the form identity plus a cubic homogeneous polynomial.^[13] This reduction leverages formal power series expansions of inverses to bound the degrees of potential counterexamples. The conjecture is also equivalent to a stronger formulation where the Jacobian determinant is nowhere zero, as non-constant but nowhere vanishing Jacobians can be adjusted via coordinate changes to yield constant ones in characteristic zero.^[13] In fields of positive characteristic p > 0, the analogue of the conjecture fails, with counterexamples existing of polynomial maps having constant non-zero Jacobian determinant but failing to be invertible. For instance, certain étale endomorphisms of affine space constructed using Frobenius actions exhibit Jacobian determinant 1 yet have fibers of arbitrary finite cardinality, violating injectivity. Over the real numbers, the Jacobian conjecture remains open.

History and Development

Origins and early formulations

In the early 20th century, algebraic geometers began examining polynomial mappings and their invertibility, particularly in the context of automorphisms of affine spaces. This interest stemmed from efforts to classify birational transformations, including those extending the classical Cremona transformations—biregular maps of projective space that are rational but not necessarily polynomial. For the affine plane \mathbb{C}^2, researchers sought to understand the group of polynomial automorphisms, which preserve the structure of the space and relate to the affine Cremona group, motivating questions about when a polynomial map is invertible via another polynomial. These explorations laid groundwork for distinguishing linear (affine) transformations, where invertibility follows from a nonzero constant Jacobian, from higher-degree cases.^[2] The conjecture itself emerged in 1939 with Ott-Heinrich Keller's seminal paper on "Ganze Cremona-Transformationen" (integral Cremona transformations), where he posed the question in the German mathematical literature. Keller, motivated by the classification of polynomial automorphisms and their connection to Cremona-like maps on affine spaces, conjectured that a polynomial endomorphism F: \mathbb{C}^n \to \mathbb{C}^n with integer coefficients is invertible by a polynomial if and only if its Jacobian determinant is a nonzero constant. He verified this for birational maps (degree 1) and low dimensions, framing it as a criterion for global invertibility based on local properties, thus linking affine transformations to broader questions in algebraic geometry. This formulation highlighted the challenge of extending affine linear invertibility to nonlinear polynomial settings.^[15]

Key contributions and researchers

In 1982, Hyman Bass, Edwin H. Connell, and David Wright established a major milestone by proving the Jacobian conjecture for quadratic polynomial maps in any number of variables over fields of characteristic zero, demonstrating that such maps are globally invertible with polynomial inverses when the Jacobian determinant is a nonzero constant. Their work also introduced a pivotal degree reduction theorem, showing that the conjecture holds for arbitrary degrees if it is true for homogeneous polynomials of degree at most three, thereby narrowing the problem significantly. This reduction further inspired Wright's conjecture, a related variant positing that the original conjecture is equivalent to its validity for cubic homogeneous maps.^[13]^[16] Arno van den Essen's 2000 monograph, Polynomial Automorphisms and the Jacobian Conjecture, serves as a definitive compilation of results accumulated by that point, synthesizing partial proofs, equivalent formulations, and historical developments while emphasizing the conjecture's implications for polynomial automorphisms. Within this text, van den Essen outlines computational and structural evidence indicating that the conjecture may fail for sufficiently large numbers of variables n, though no explicit counterexamples have been constructed and the claim remains unproven.^[17] Subsequent advancements in reductions have been driven by researchers such as Michiel de Bondt, who, in joint work with van den Essen, established in 2005 that the conjecture suffices to be proven for symmetric homogeneous polynomial maps of a specific form, further simplifying the general case.^[18] Post-2018 developments have leveraged computational techniques to verify the conjecture in low dimensions, particularly for n=2, where results confirm its validity up to polynomial degrees exceeding 100; a 2025 extension using algorithmic methods has pushed this verification to degree 104, ruling out counterexamples in these regimes without yielding a general proof.^[19] The Jacobian conjecture persists as an open problem as of 2025, with no counterexamples identified over algebraically closed fields of characteristic zero, yet recent discussions, including those on MathOverflow, have prompted inquiries into its possible undecidability, framing it as a Π⁰₂ statement in the arithmetic hierarchy that might resist resolution by standard mathematical methods.^[5]

Algebraic and geometric methods

One prominent algebraic approach to the Jacobian conjecture involves analyzing the structure of the automorphism group of the affine space \mathbb{A}^n, denoted GA_n(k), over an algebraically closed field k of characteristic zero. The conjecture asserts that a polynomial map F: k^n \to k^n with constant non-zero Jacobian determinant is an automorphism, meaning it belongs to GA_n(k), the group generated by affine transformations and elementary automorphisms (linear plus shear maps). Seminal work reduced the problem by showing that if F has degree d > 1, then the conjecture holds for F if and only if it holds for a related map of degree 3, using formal power series expansions of the inverse and bounding the nilpotence index of the Jacobian matrix at the origin. This approach aims to prove that such maps are tame automorphisms, i.e., compositions of affine and elementary ones, leveraging the known structure of GA_n(k) in low dimensions via the Jung-van der Kulk theorem.^[2] Geometric methods interpret the polynomial map F: \mathbb{A}^n \to \mathbb{A}^n via its graph in \mathbb{A}^n \times \mathbb{A}^n, compactified to a projective variety after resolving indeterminacies through blow-ups. In dimension n=2, the graph becomes a surface X with projections \pi: X \to \mathbb{P}^2 and \phi: X \to \mathbb{P}^2, where the exceptional locus \pi^{-1}(\infty) forms a tree of smooth rational curves. The constant Jacobian condition implies birationality of \phi, and ampleness of the pullback sheaf \phi^* \mathcal{O}_{\mathbb{P}^2}(1) on X would confirm this, as ample line bundles on rational surfaces ensure the map is a morphism with polynomial inverse. Invariants like the canonical divisor labels and determinants of intersection forms on exceptional curves bound the number of blow-ups (at most N for degree d), with negative labels identifying lines at infinity. This birational framework has partial success in low dimensions but faces challenges in higher n due to singularity resolution.^[20] Combinatorial attempts, particularly those by David Wright, utilize rewriting systems and tree structures to reduce the conjecture to the case of tame automorphisms. Wright's diamond lemma, adapted from ring theory, provides a confluence condition for reducing expressions in the free group of automorphisms, allowing decomposition into elementary factors when the Jacobian matrix J(H) for F = x - H is nilpotent of low index (e.g., J(H)^3 = 0). The Bass-Connell-Wright tree inversion formula expresses the formal inverse as a linear combination over rooted trees, and the lemma ensures unique normal forms, proving the conjecture for symmetric homogeneous cubics of degree up to 4 in n \leq 4 variables by showing higher-weight trees vanish. This yields partial success, improving degree bounds to 6 for binary trees with at least 7 leaves, but fails for general nilpotence indices greater than 3.^[21]

Connections to other areas of mathematics

The Jacobian conjecture exhibits connections to dynamical systems through polynomial maps with constant Jacobian determinants, which relate to the classification of isochronous centers in Hamiltonian systems. Specifically, in the two-dimensional real case, determining whether certain polynomial Hamiltonian systems possess isochronous centers—singular points surrounded by periodic orbits of equal period—is equivalent to resolving the conjecture, as such centers imply global invertibility of the associated maps. This linkage arises because constant Jacobian maps preserve volume and exhibit symmetries akin to integrable Hamiltonian flows, where the conjecture's injectivity condition mirrors the global stability required for isochronous behavior.^[22] Reformulations of the conjecture also intersect with partial differential equation (PDE) integrability, particularly via associated nonlinear PDEs derived from the Jacobian condition. One approach separates the Jacobian equation into subequations for a multiply parametrized family of polynomial maps, enabling systematic integration that yields broad classes of maps satisfying the conjecture in arbitrary dimensions and degrees. This PDE perspective highlights integrability as a pathway to resolution, with recent work identifying explicit solutions that confirm the conjecture for specific families while suggesting broader applicability.^[23] Arithmetic variants extend the conjecture to p-adic settings through the Tate-Jacobian conjecture, which posits that for Tate algebras over complete discrete valuation rings like \mathbb{C}_p, a polynomial map with Jacobian determinant in the units has an inverse within the Tate algebra. This holds for all but finitely many primes p under certain topological conditions, linking the original conjecture to arithmetic geometry over non-archimedean fields. Furthermore, the conjecture belongs to the \Pi^0_1 class in the arithmetic hierarchy, meaning it is a universal arithmetic statement; while undecidable in general due to its openness, it is algorithmically decidable for fixed polynomial degrees via finite computations.^[24]^[5] In combinatorics, the conjecture reduces to symmetric homogeneous cases that admit combinatorial interpretations, such as inversion formulas tied to structures like the Grossman-Larson algebra, providing a framework for verifying injectivity through counting and algebraic relations. Computational bounds arise from this view, enabling algorithmic checks for low-degree instances, though exponential growth in complexity underscores the challenge for higher dimensions. Potential ties to Hilbert's 13th problem emerge indirectly via questions of polynomial decompositions, but remain exploratory without direct equivalence.^[18] Weak variants, such as Kaliman's conjecture on real injectivity, have seen advancements using the Hurwitz formula for analyzing morphisms on Riemann surfaces. A 2024 improvement establishes that if a polynomial map F: \mathbb{C}^2 \to \mathbb{C}^2 has constant nonzero Jacobian and infinitely many linear combinations aF_1 + bF_2 + c are irreducible, then F is invertible, implying global injectivity over the reals via resolution of singularities. This refines the real-analytic case and offers tools for partial resolutions in characteristic zero.^[25]

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