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In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse.

(two variables), also called the plane Jacobian conjecture or planar Jacobian conjecture, is the only case that remains unresolved . and linear polynomial functions are invertible.

On July 19, 2026, Levent Alpöge presented an explicit counterexample in three variables which he credited to Claude Fable 5, which disproves the conjecture for . The correctness of the counterexample is easy to verify with any computer algebra system. It has not been revealed, however, how it was found. Nevertheless, it led some mathematicians to elaborate on the mathematical reasons and the implications of the existence of the counterexample.

History

The Jacobian conjecture was originally formulated without being named in two dimensions by Ludwig Kraus in 1884, who gave a flawed proof in the same paper. dimensions was formulated by Ott-Heinrich Keller in 1939 for the case of polynomials with integer coefficients. Arno van den Essen has claimed that Keller only talked about the two-dimensional case;-dimensional case. For nearly a century, Keller was considered to be the first person to formulate the conjecture, until a 2025 search of the zbMATH database revealed that the two-dimensional case had already been stated by Kraus.

The conjecture was unnamed in either of Kraus's or Keller's original papers. It is unclear who coined the name "Jacobian conjecture", but this was because it involves the Jacobian determinant, itself named after the German mathematician Carl Gustav Jacob Jacobi-dimensional conjecture. later recalled that, after Oscar Zariski pointed out at a seminar at Purdue University in the late 1960s that the assertion remained unproved, "we decided to call it the Jacobian Conjecture". Alexander Borisov later attributed the coinage specifically to Shreeram Abhyankar, whose 1977 lecture notes treated the two-dimensional problem and presented results that he had obtained in 1970–71.

The conjecture, also called Jacobian problem, was subsequently widely publicized by Abhyankar as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century.

According to Alexander Borisov, the conjecture in two dimensions has been especially well studied in the literature on the conjecture. Historically, some mathematicians, such as Shreeram Abhyankar, Tzuong-Tsieng Moh, and Yitang Zhang, have used the term Jacobian conjecture or Jacobian problem to refer to only the conjecture in two dimensions. There have been a large number of false proofs of the conjecture in two dimensions, some of them published. Arno van den Essen in 1997, Tzuong-Tsieng Moh in 1998, and Edward Formanek in 2011 all hypothesized that the conjecture could be true in two dimensions and false in the general case. After the counterexample in three dimensions was discovered in 2026 only the conjecture in two dimensions remains open.

Formulation of the conjecture

with coefficients in a field. Then we define a vector-valued function

arising in this way is called a polynomial mapping.

The Jacobian determinant, is defined as the determinant Jacobian matrix consisting of the partial derivatives:

It follows from the multivariable chain rule has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a smooth local to itself) and has constant Jacobian 1. This means intuitively that it bends the plane around like a rubber sheet without locally expanding or compressing area, so it cannot create pinches or other kinds of degeneration locally: a small disc around each point is sent one-to-one onto a small domain in the image., and so causes different parts of the plane to overlap in the image. The analog of the Jacobian conjecture is therefore false for analytic maps; the conjecture is hard because the map is required to be a polynomial.

an example of Pierre Fatou and Ludwig Bieberbach.

Results

General results and reductions

using the Lefschetz principle. is injective, it can be shown to already be bijective with a regular inverse (cf. the Ax–Grothendieck theorem).

. In light of the counterexample, the positive partial results now describe conditions that any counterexample must violate, while the reductions show that counterexamples of quite special forms must exist.

is simply a set of functions linear in the variables, because then the inverse will also be a set of linear functions. However, unlike the 1-dimensional case, in two or more dimensions, there exist nonlinear polynomial maps with constant Jacobian determinant and a polynomial inverse. A simple quadratic example is given by

so that the Jacobian determinant is

In this case the inverse exists as the polynomials

Stuart Sui-Sheng Wang proved the Jacobian conjecture for polynomials of degree 2, so any counterexample must have degree at least 3. Hyman Bass is either zero or a homogeneous cubic. are cubes of homogeneous linear polynomials. These reductions introduce additional variables and therefore do not preserve a fixed dimension.

Edwin Connell and Lou van den Dries proved that if the Jacobian conjecture is false, then it has a counterexample with integer coefficients and Jacobian determinant 1.

denote the polynomial ring. Keller (1939) proved the birational is a Galois extension was proved by Andrew Campbell for complex maps and in general by Michael Razar and, independently, by David Wright. No counterexample can therefore be birational or define a Galois extension; consistent with this, the 2026 counterexample is generically three-to-one.

Michiel de Bondt and Arno van den Essen and Ludwik Drużkowski; no counterexample of this more restricted form is therefore possible, so the cubic-linear and symmetric reductions cannot be combined.

The strong real Jacobian conjecture was the assertion that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. Sergey Pinchuk constructed a counterexample to the strong real Jacobian conjecture of total degree 35. Because Pinchuk's maps have nonconstant Jacobian determinant, they did not disprove the Jacobian conjecture itself.

, the Dixmier conjecture states that every endomorphism-th Weyl algebra over a field of characteristic zero is an automorphism-dimensional Jacobian conjecture. Conversely, Yoshifumi Tsuchimoto and, independently, Alexei Belov-Kanel and Maxim Kontsevich.

Kossivi Adjamagbo and Arno van den Essen gave a self-contained algebraic proof and established the more precise chain of implications

variables. Consequently, the all-dimensional forms of the Jacobian, Poisson, and Dixmier conjectures are equivalent.

) but it has no inverse function. However, does not divide the degree of the field extension; this is called the separable Jacobian conjecture because Adjamagbo's condition guarantees that the field extension is separable., and in another 2026 preprint, Romy Mondello claimed to have produced a counterexample in two dimensions.

In two dimensions

The conjecture remains open in two dimensions . Much of the study of the two-dimensional conjecture concerns the geometry of a hypothetical counterexample at infinity. In two variables, a pair of polynomials is called a Jacobian pair if its Jacobian determinant is a nonzero constant. Shreeram Abhyankar proved that each component of a Jacobian pair has at most two points at infinity. He also showed that the planar conjecture is equivalent to each of several assertions about all Jacobian pairs, including that each component has only one point at infinity, that the Newton polygon of each component is a triangle, and that one of the two component degrees divides the other. Using characteristic sequences and approximate roots, Abhyankar proved the conjecture in the case of two characteristic pairs and later sharpened this to what he called the "two plus epsilon characteristic pairs" case. His subsequent work developed the more general "three minus epsilon characteristic pairs" setting.

, and obtained further restrictions on the possible shape of a counterexample. Leonid Makar-Limanov later introduced a Newton polyhedron associated with a minimal counterexample, obtaining a sharper estimate for its geometric degree and a new proof of Abhyankar's result for two characteristic pairs.

More recent geometric approaches study hypothetical Jacobian pairs through compactifications of the affine plane. Alexander Borisov formulated the resulting constraints on the maps between the Picard groups of suitable compactifications as an essentially combinatorial problem and described several possible configurations, which he called "frameworks". A framework is not itself a counterexample: each gives a system of equations whose solution would produce a Jacobian pair.

A complementary line of work concerns the possible covering behavior of a counterexample. In 1971, Anatoli Vitushkin constructed a three-sheeted branched covering whose regular part is contractible and whose bifurcation set is homeomorphic to a plane. cannot be a nonzero constant., while Nguyen Van Chau proved that the Vitushkin covering cannot occur in the polynomial case. with nonzero constant Jacobian. One of his examples has a pole along a complex line, but on the complement of that line it defines a two-sheeted étale cover; in particular, it is generically two-to-one, but is not a polynomial counterexample.

A separate computational line has sought to exclude counterexamples of progressively higher degree. Tzuong-Tsieng Moh's 1983 computer-assisted argument, with its algorithm subsequently revised by Lih-Chung Wang in 2005, verified it for polynomials of degree at most 100. This bound was increased to 104 by Thuy Nguyen in 2025. In a 2022 preprint, Jorge Alberto Guccione, Juan José Guccione, Rodrigo Horruitiner, and Christian Valqui claimed that this bound can be increased to 124 except for the possible degree pair (72,108).

Counterexample for n > 2

On July 19, 2026, mathematician and Anthropic employee Levent Alpöge presented an explicit counterexample to the conjecture in three-dimensional space which he credited to Claude Fable 5 AI model. According to Abhishek Saha of the Queen Mary University of London, the counterexample is simple to verify in itself, but how Alpöge and Fable exactly arrived at it is unclear.

is injective, then it is a polynomial automorphism. is a coordinate polynomial, and is therefore irreducible. cannot be injective.

variables. The same holds for any other counterexample in dimension 3.

The day after Alpöge found the counterexample, a geometric reformulation was announced by Andy Jiang, a doctoral student in mathematics at the University of Michigan, who credited it to "GPT".

Terence Tao, a mathematician from the University of California, Los Angeles (UCLA), discussed this geometric explanation of the counterexample using multiplication of binary forms. A generic binary cubic has three linear factors, and hence three ways to express it as the product of a distinguished linear factor and the remaining quadratic factor. After imposing a resultant normalization to remove the scaling ambiguity and restricting to a particular affine slice of the space of binary cubics, Tao obtained an étale. This cubic factorization construction is thus similar to the earlier quadratic example by Vitushkin.