Jacobian conjecture

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.^[5][6][1] Later, the modern version of the Jacobian conjecture in ${\displaystyle n}$ dimensions was formulated by Ott-Heinrich Keller in 1939 for the case of polynomials with integer coefficients.^[7] Arno van den Essen has claimed that Keller only talked about the two-dimensional case;^[8] however, Keller in fact did talk about the general ${\displaystyle n}$-dimensional case.^[7] For nearly a century, Keller was considered to be the first person to formulate the two-dimensional case, until a 2025 search of the zbMATH database revealed that the two-dimensional case had already been stated by Kraus.^[1]

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 ${\displaystyle n}$-dimensional conjecture.^[9] Tzuong-Tsieng Moh [zh; de] 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".^[10] Alexander Borisov later attributed the coinage specifically to Shreeram Abhyankar,^[11] whose 1977 lecture notes treated the two-dimensional problem and presented results that he had obtained in 1970–71.^[12]

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.^[12][13] The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century.^[14]

According to Alexander Borisov, the conjecture in two dimensions has been especially well studied in the literature on the conjecture.^[15] 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.^{[12][16][17][18][19]} There have been a large number of false proofs of the conjecture in two dimensions,^[20][21] some of them published.^[22][23][18] 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.^[24][10][25] After the counterexample in three dimensions, credited to Claude Fable 5, was discovered in 2026, only the conjecture in two dimensions remains open.^[3]

Let ${\displaystyle n>1}$ be a fixed integer and consider polynomials ${\displaystyle f_{1},\dots ,f_{n}}$ in variables ${\displaystyle X_{1},\dots ,X_{n}}$ with coefficients in a field ${\displaystyle \mathbb {K} }$. Then we define a vector-valued function ${\displaystyle F:\mathbb {K} ^{n}\to \mathbb {K} ^{n}}$ by setting: $${\displaystyle F(X_{1},\dots ,X_{n})=(f_{1}(X_{1},\dots ,X_{n}),\dots ,f_{n}(X_{1},\dots ,X_{n}))}$$

Any map ${\displaystyle F:\mathbb {K} ^{n}\to \mathbb {K} ^{n}}$ arising in this way is called a polynomial mapping.

The Jacobian determinant of ${\displaystyle F}$, denoted by ${\displaystyle J_{F}}$, is defined as the determinant of the ${\displaystyle n\times n}$ Jacobian matrix consisting of the partial derivatives of ${\displaystyle f_{i}}$ with respect to ${\displaystyle X_{j}}$:

${\displaystyle J_{F}=\left|{\begin{matrix}{\frac {\partial f_{1}}{\partial X_{1}}}&\cdots &{\frac {\partial f_{1}}{\partial X_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{n}}{\partial X_{1}}}&\cdots &{\frac {\partial f_{n}}{\partial X_{n}}}\end{matrix}}\right|,}$

then ${\displaystyle J_{F}}$ is itself a polynomial function of the ${\displaystyle n}$ variables ${\displaystyle X_{1},\dots ,X_{n}}$.

It follows from the multivariable chain rule that if ${\displaystyle F}$ has a polynomial inverse function ${\displaystyle G:\mathbb {K} ^{n}\to \mathbb {K} ^{n}}$, then ${\displaystyle J_{F}}$ has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

Jacobian conjecture: Let ${\displaystyle \mathbb {K} }$ have characteristic ${\displaystyle 0}$. If ${\displaystyle J_{F}}$ is a non-zero constant, then ${\displaystyle F}$ has an inverse function ${\displaystyle G:\mathbb {K} ^{n}\to \mathbb {K} ^{n}}$ that is regular, meaning its components are polynomials.

The condition ${\displaystyle J_{F}\neq 0}$ 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 ${\displaystyle F}$ exists at every point where ${\displaystyle J_{F}}$ is non-zero. This means there is a neighborhood of each such point that is mapped bijectively onto its image. For example, the map ${\displaystyle x\mapsto x+x^{3}}$ 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 ${\displaystyle F(x,y)=(e^{x}\cos(e^{-2x}y),e^{x}\sin(e^{-2x}y)}$ maps ${\displaystyle \mathbb {R} ^{2}}$ to itself (or ${\displaystyle \mathbb {C} ^{2}}$ 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.^[26][27] Nevertheless, the function maps all points ${\displaystyle (x,y+2\pi e^{2x}k),k\in \mathbb {Z} }$ to the same point ${\displaystyle F(x,y)}$, and so causes different parts of the plane to overlap in the image.^[28] 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.^[29]

The case of polynomials over a field ${\displaystyle \mathbb {K} }$ of characteristic zero can be reduced to ${\displaystyle \mathbb {K} =\mathbb {C} }$ using the Lefschetz principle.^[20] Further, if ${\displaystyle F}$ is injective, it can be shown to already be bijective with a regular inverse^[30] (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 ${\displaystyle n>2}$. 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 ${\displaystyle F}$ 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

${\displaystyle u=x^{2}+y+x}$

${\displaystyle v=x^{2}+y}$

so that the Jacobian determinant is

${\displaystyle J_{F}=\left|{\begin{matrix}1+2x&1\\2x&1\end{matrix}}\right|=(1+2x)(1)-(1)2x=1.}$

In this case the inverse exists as the polynomials

${\displaystyle x=u-v}$

${\displaystyle y=v-(u-v)^{2}.}$

Stuart Sui-Sheng Wang proved the Jacobian conjecture for polynomials of degree 2,^[31] 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 ${\displaystyle F=(X_{1}+H_{1},\dots ,X_{n}+H_{n})}$, where each ${\displaystyle H_{i}}$ is either zero or a homogeneous cubic.^[32] Ludwik Drużkowski showed that one may further assume that the map is of cubic linear type, meaning that the nonzero ${\displaystyle H_{i}}$ are cubes of homogeneous linear polynomials.^[33]

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.^[34]

Let ${\displaystyle \mathbb {K} [X]}$ denote the polynomial ring ${\displaystyle \mathbb {K} [X_{1},\dots ,X_{n}]}$ and ${\displaystyle \mathbb {K} [F]}$ denote the ${\displaystyle \mathbb {K} }$-subalgebra generated by ${\displaystyle f_{1},\dots ,f_{n}}$. For a given ${\displaystyle F}$, the Jacobian condition implies invertibility if and only if ${\displaystyle \mathbb {K} [X]=\mathbb {K} [F]}$. Keller (1939) proved the birational case, that is, where the two fields ${\displaystyle \mathbb {K} (X)}$ and ${\displaystyle \mathbb {K} (F)}$ are equal. The case where ${\displaystyle \mathbb {K} (X)}$ is a Galois extension of ${\displaystyle \mathbb {K} (F)}$ was proved by Andrew Campbell for complex maps^[35] and in general by Michael Razar^[36] and, independently, by David Wright.^[22] No counterexample can therefore be birational or define a Galois extension; consistent with this, the 2026 counterexample is generically three-to-one.^[4]

Michiel de Bondt and Arno van den Essen^[37][38] and Ludwik Drużkowski^[39] 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 ${\displaystyle 0}$; 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.^[40] Because Pinchuk's maps have nonconstant Jacobian determinant, they did not disprove the Jacobian conjecture itself.

The Dixmier conjecture, which asserted that every endomorphism of a Weyl algebra is an automorphism, implies the Jacobian conjecture in the corresponding dimension.^[32] Conversely, it was shown by Yoshifumi Tsuchimoto^[41] and independently by Alexei Belov-Kanel and Maxim Kontsevich^[42] that the Jacobian conjecture for ${\displaystyle 2n}$ variables implies the Dixmier conjecture in ${\displaystyle n}$ dimensions. A self-contained and purely algebraic proof of the last implication was given by Kossivi Adjamagbo and Arno van den Essen,^[43] who also proved in the same paper that these two conjectures are equivalent to the Poisson conjecture, that every endomorphism of the n-th complex Poisson algebra is an automorphism. In consequence of the 2026 counterexample, the Dixmier and Poisson conjectures are false in every dimension ${\displaystyle n>2}$, while, as with the two-variable Jacobian conjecture itself, the case ${\displaystyle n=2}$ remains open.

The obvious analogue of the Jacobian conjecture fails if ${\displaystyle \mathbb {K} }$ has characteristic ${\displaystyle p>0}$ even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least ${\displaystyle 2}$. The polynomial ${\displaystyle x-x^{p}}$ has derivative ${\displaystyle 1-px^{p-1}}$, which is ${\displaystyle 1}$ (because ${\displaystyle p=0}$ in characteristic ${\displaystyle p}$) but it has no inverse function. However, Kossivi Adjamagbo [ht] suggested extending the Jacobian conjecture to characteristic ${\displaystyle p>0}$ by adding the hypothesis that ${\displaystyle p}$ does not divide the degree of the field extension ${\displaystyle \mathbb {K} (X)/\mathbb {K} (F)}$;^[44] this is called the separable Jacobian conjecture because Adjamagbo's condition guarantees that the field extension is separable.^[45][46] For the separable Jacobian conjecture in characteristic two, in a 2026 preprint, Irit Huq-Kuruvilla claimed to have produced a counterexample in dimensions ${\displaystyle n\geq 3}$,^[45] and in another 2026 preprint, Romy Mondello claimed to have produced a counterexample in two dimensions.^[46]

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