Jacobian conjecture

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller. It was later named and widely publicised by Shreeram Abhyankar, as an example of a question in the area of algebraic geometry that requires little beyond a knowledge of calculus to state.

The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. As of 2015, there are no plausible claims to have proved it.

The Jacobian determinant[edit]

Let N > 1 be a fixed integer and consider the polynomials f1, ..., fN in variables X1, ..., XN with coefficients in the complex numbers k (in fact, this case is equivalent to the Jacobian conjecture for any characteristic zero field k). Then we define a vector-valued function F: kNkN by setting:

F(c1, ..., cN) = (f1(c1, ...,cN),..., fN(c1,...,cN))

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:

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 JF is itself a polynomial function of the N variables X1, …, XN.

Formulation of the conjecture[edit]

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 local inverse function to F exists at any point where JF is non-zero. However k is algebraically closed so JF as a polynomial will be zero for some complex values of X1, …, XN unless it is a non-zero constant function. It holds true that:

Proposition: If F has an inverse function G: kNkN, then JF is a non-zero constant.

The conjecture is the following converse:

Jacobian conjecture: If JF is a non-zero constant, then F has an inverse function G: kNkN, and G is regular (in the sense that its components are given by polynomial expressions).

The obvious analogue of the Jacobian conjecture fails in characteristic p>0 even for 1 variable, as the polynomial x+xp has derivative 1 but no polynomial inverse function.


Wang (1980) proved the Jacobian conjecture for polynomials of degree 2, and Bass, Connell & Wright (1982) showed that the general case follows from the special case where the polynomials are of degree 3, more particularly, of the form F=(X1+H1,...,Xn+Hn), where each Hi is either zero or a homogeneous cubic. In this case, invertibility of the Jacobian is equivalent to the Jacobian matrix being unipotent. Moh (1983) checked the conjecture for polynomials of degree at most 100 in 2 variables. De Bondt and Van den Essen (2005) showed that it is even enough to prove the Jacobian Conjecture in the cases where the Jacobian matrix is symmetric.

P.K. Adjamagbo and A. van den Essen (2007) and Alexei Belov-Kanel and Maxim Kontsevich (2007) showed that the Jacobian conjecture is equivalent to the Dixmier conjecture.

The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. It was disproved by Sergey Pinchuk (1994).


External links[edit]