Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse. It was first conjectured in 1939 by Ott-Heinrich Keller, and widely publicized by Shreeram 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The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors. since 2018Template:Dated maintenance category (articles), it has not been proven, even for the two-variable case. Van den Essen provides evidence that the conjecture may be false for large numbers of variables.

The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century.

The Jacobian determinant

Let N > 1 be a fixed integer and consider polynomials f₁, ..., f_N in variables X₁, ..., X_N with coefficients in a field k. Then we define a vector-valued function F: k^N → k^N by setting:

F(X₁, ..., X_N) = (f₁(X₁, ...,X_N),..., f_N(X₁,...,X_N)).

Any map F: k^N → k^N arising in this way is called a polynomial mapping.

The Jacobian determinant of F, denoted by J_F, is defined as the determinant of the N × N Jacobian matrix consisting of the partial derivatives of f_i with respect to X_j:

$J_F=\left|\begin{array}{ccc}\frac{\partial f_1}{\partial X_1}& \cdots & \frac{\partial f_1}{\partial X_N}\\ ⋮& \ddots & ⋮\\ \frac{\partial f_N}{\partial X_1}& \cdots & \frac{\partial f_N}{\partial X_N}\end{array}\right|,$

then J_F is itself a polynomial function of the N variables X₁, ..., X_N.

Formulation of the conjecture

It follows from the multivariable chain rule that if F has a polynomial inverse function G: k^N → k^N, then J_F has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

Jacobian conjecture: Let k have characteristic 0. If J_F is a non-zero constant, then F has an inverse function G: k^N → k^N which is regular, meaning its components are polynomials.

According to van den Essen, the problem was first conjectured by Keller in 1939 for the limited case of two variables and integer coefficients.

The obvious analogue of the Jacobian conjecture fails if k has characteristic p > 0 even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least 2. The polynomial x − x^p has derivative 1 − p x^p−1 which is 1 (because px is 0) but it has no inverse function. However, Template:Ill suggested extending the Jacobian conjecture to characteristic p > 0 by adding the hypothesis that p does not divide the degree of the field extension k(X) / k(F).

The existence of a polynomial inverse is obvious if F is simply a set of functions linear in the variables, because then the inverse will also be a set of linear functions. A simple non-linear example is given by

$u=x^2+y+x$

$v=x^2+y$

so that the Jacobian determinant is

$J_F=\left|\begin{array}{cc}1+2x& 1\\ 2x& 1\end{array}\right|=(1+2x)(1)-(1)2x=1.$

In this case the inverse exists as the polynomials

$x=u-v$

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

But if we modify F slightly, to

$u=2x^2+y$

$v=x^2+y$

then the determinant is

$J_F=\left|\begin{array}{cc}4x& 1\\ 2x& 1\end{array}\right|=(4x)(1)-2x(1)=2x,$

which is not constant, and the Jacobian conjecture does not apply. The function has an inverse function when we work over the reals and assume $x>0$:

$x=\sqrt{u-v}$

$y=2v-u,$

However, this expression for x is not a polynomial. If instead we work over the complex numbers the square root is multivalued.

The condition J_F ≠ 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 F exists at every point where J_F is non-zero. For example, the map x → x + x³ has a smooth global inverse, but the inverse is not polynomial.

Results

Stuart Sui-Sheng Wang proved the Jacobian conjecture for polynomials of degree 2. 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 F = (X₁ + H₁, ..., X_n + H_n), where each H_i 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 H_i are cubes of homogeneous linear polynomials. These reductions introduce additional variables and so are not available for fixed N.

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. In consequence, the Jacobian conjecture is true either for all fields of characteristic 0 or for none. For fixed dimension N, it is true if it holds for at least one algebraically closed field of characteristic 0.

Let k[X] denote the polynomial ring k[X₁, ..., X_n] and k[F] denote the k-subalgebra generated by f₁, ..., f_n. For a given F, the Jacobian conjecture is true if, and only if, k[X] = k[F]. Keller (1939) proved the birational case, that is, where the two fields k(X) and k(F) are equal. The case where k(X) is a Galois extension of k(F) was proved by Andrew Campbell for complex maps and in general by Michael Razar and, independently, by David Wright. Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.

Michiel de Bondt and Arno van den Essen and Ludwik Drużkowski independently showed that it is enough to prove the Jacobian Conjecture for complex maps of cubic homogeneous type with a symmetric Jacobian matrix, and further showed that the conjecture holds for maps of cubic linear type with a symmetric Jacobian matrix, over any field of characteristic 0.

The strong real Jacobian conjecture was 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 two variable counterexamples of total degree 35 and higher.

It is well known that the Dixmier conjecture, that every endomorphism of a Weyl algebra is an automorphism, implies the Jacobian conjecture. Conversely, it is shown by Yoshifumi Tsuchimoto and independently by Alexei Belov-Kanel and Maxim Kontsevich that the Jacobian conjecture for 2N variables implies the Dixmier conjecture in N dimensions. A self-contained and purely algebraic proof of the last implication is also given by Kossivi Adjamagbo and Arno van den Essen who also proved in the same paper that these two conjectures are equivalent to the Poisson conjecture, that every endomorphism of the Nth complex Poisson algebra is an automorphism.

See also

• List of unsolved problems in mathematics

External links

• Web page of Tzuong-Tsieng Moh on the conjecture