In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an -dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse.
The case (two variables), also called the plane Jacobian conjecture or planar Jacobian conjecture, is the only case that remains unresolved . The case is trivially true, since the derivative of a polynomial is a nonzero constant only if the degree of the polynomial is and linear polynomial functions are invertible.
On July 19, 2026, Levent Alpöge presented an explicit counterexample in three variables which he credited to the large language model Claude Fable 5, disproving 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.
For fields of positive characteristic, the usual Jacobian conjecture is false in one dimension, so the separable Jacobian conjecture, also called Adjamagbo's Jacobian conjecture, adds a separability premise to make the conjecture true in one dimension. The separable Jacobian conjecture has been proven false for all fields of positive characteristic in all dimensions , with the separable Jacobian conjecture for fields of characteristic zero coinciding with the usual Jacobian conjecture.
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. Later, the modern version of the Jacobian conjecture in 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; however, Keller in fact did talk about the general -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. The earliest known published use of the name occurs in Masayoshi Miyanishi's 1973 paper, where it refers to the -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 Let be a fixed integer and consider polynomials in variables with coefficients in a field . Then we define a vector-valued function by setting:
Any map arising in this way is called a polynomial mapping.
The Jacobian determinant of , denoted by , is defined as the determinant of the Jacobian matrix consisting of the partial derivatives of with respect to :
then is itself a polynomial function of the variables .
It follows from the multivariable chain rule that if has a polynomial inverse function , then has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:
The condition is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a smooth local inverse function to exists at every point where is non-zero. This means there is a neighborhood of each such point that is mapped bijectively onto its image. For example, the real function has a smooth global inverse, but the inverse is not polynomial. A differentiable function with a non-zero Jacobian is locally invertible in higher dimensions as well. For example, the function maps to itself (or 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. Nevertheless, the function maps all points to the same point , 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.
Apart from injectivity, the Jacobian conjecture is distinguished from analytic mappings because an injective polynomial mapping of to itself is automatically surjective. This is not true for analytic mappings, and for an example of Pierre Fatou and Ludwig Bieberbach shows that the image of an injective analytic mapping can exclude a nonempty open subset of .
Results General results and reductions The case of polynomials over a field of characteristic zero can be reduced to using the Lefschetz principle. Further, if is injective, it can be shown to already be bijective with a regular inverse (cf. the Ax–Grothendieck theorem).
Many special cases and reductions of the Jacobian conjecture were established in the decades before it was disproved in 2026 for . 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.
The existence of a polynomial inverse is obvious if 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, Edwin Connell, and David Wright showed that the general case follows from the special case where the polynomials are of degree 3, or even more specifically, of cubic homogeneous type, meaning of the form , where each is either zero or a homogeneous cubic. Ludwik Drużkowski showed that one may further assume that the map is of cubic linear type, meaning that the nonzero 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.
Let denote the polynomial ring and denote the -subalgebra generated by . For a given , the Jacobian condition implies invertibility if and only if . Keller (1939) proved the birational case, that is, where the two fields and are equal. The case where is a Galois extension of 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 independently showed that the general case of the conjecture is equivalent to the special case of complex maps of cubic homogeneous type with a symmetric Jacobian matrix, so counterexamples of this form must also exist. They further showed that the conjecture holds for maps of cubic linear type with a symmetric Jacobian matrix, over any field of characteristic ; 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.
For , the Dixmier conjecture states that every endomorphism of the -th Weyl algebra over a field of characteristic zero is an automorphism. For each , the Dixmier conjecture for implies the -dimensional Jacobian conjecture. Conversely, Yoshifumi Tsuchimoto and, independently, Alexei Belov-Kanel and Maxim Kontsevich showed that the Jacobian conjecture in variables implies the Dixmier conjecture for .
Kossivi Adjamagbo and Arno van den Essen gave a self-contained algebraic proof and established the more precise chain of implications
where denotes the Poisson conjecture for the canonical complex Poisson algebra in variables. Consequently, the all-dimensional forms of the Jacobian, Poisson, and Dixmier conjectures are equivalent.
Since the 2026 counterexample disproves for every , the above implications show that and are also false for every , while both conjectures remain open for .
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.
Another line of work has progressively restricted the possible degree pairs and Newton polygons of a counterexample. Raymond C. Heitmann showed that any counterexample must satisfy