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 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 attempted proofs that turned out to contain subtle errors. As of 2018, there are no plausible claims to have proved it. Even the two-variable case has resisted all efforts. There are currently no known compelling reasons for believing the conjecture to be true, and according to van den Essen there are some suspicions that the conjecture is in fact false for large numbers of variables (indeed, there is equally also no compelling evidence to support these suspicions). 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, ..., fN in variables X1, ..., XN with coefficients in a field k. Then we define a vector-valued function F: kN → kN by setting:
F(X1, ..., XN) = (f1(X1, ...,XN),..., fN(X1,...,XN)).
Any map F: kN → kN arising in this way is called a polynomial mapping.
The Jacobian determinant of F, denoted by JF, is defined as the determinant of the N × N Jacobian matrix consisting of the partial derivatives of fi with respect to Xj:
then JF is itself a polynomial function of the N variables X1, ..., XN.
Formulation of the conjecture
It follows from the multivariable chain rule that if F has a polynomial inverse function G: kN → kN, then JF 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 JF is a non-zero constant, then F has an inverse function G: kN → kN 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 has derivative which is 1 (because px is 0) but it has no inverse function. However, suggested extending the Jacobian conjecture to characteristic by adding the hypothesis that p does not divide the degree of the field extension .
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 would be if
in which case the Jacobian determinant is
In this case the inverse exists as the polynomials
But if we modify F just slightly, as
then the determinant is
so not constant, and the Jacobian conjecture does not apply. The function still has an inverse:
but the expression for x is not a polynomial.
The condition JF ≠ 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 JF 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, ..., Xn + Hn), where each Hi 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 Hi are cubes of homogeneous linear polynomials. It seems that Drużkowski's reduction is one most promising way to go forward. These reductions introduce additional variables and so are not available for fixed N.