Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller. It was widely publicized 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 2016, there are no plausible claims to have proved it. Even the two variable case has resisted all efforts. There are no known compelling reasons for believing it to be true, and according to van den Essen (1997) there are some suspicions that the conjecture is in fact 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, ..., fN in variables X1, ..., XN with coefficients in a field k. Then we define a vector-valued function F: kN → kN 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:

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: If JF is a non-zero constant and k has characteristic 0, then F has an inverse function G: kN → kN, and G is regular (in the sense that its components are polynomials).

The obvious analogue of the Jacobian conjecture fails if k has characteristic p > 0 even for 1 variable. The characteristic of a field must be prime, so it is at least 2. The polynomial x − xp has derivative 1 − px xp−2 which is 1 (because px is 0) but it has no inverse function. However, Adjamagbo (1995) 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).

For real maps, 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

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, 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. Drużkowski (1983) 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. These reductions introduce additional variables and so are not available for fixed N.

Connell & van den Dries (1983) 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 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, ..., Xn] and k[F] denote the k-subalgebra generated by f1, ..., fn. 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 Campbell (1973) for complex maps and in general by Razar (1979) and, independently, Wright (1981). Adjamagbo (1995) 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). Moh (1983) checked the conjecture for polynomials of degree at most 100 in two variables.

de Bondt, van den Essen & 2005, 2005 and Drużkowski (2005) 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 (1994) constructed two variable counterexamples of total degree 25 and higher.

It is well-known that the Dixmier conjecture implies the Jacobian conjecture (see Bass et al. 1982). Conversely, it is shown by Yoshifumi Tsuchimoto (2005),[1] and independently by Alexei Belov-Kanel and Maxim Kontsevich (2007), that the Jacobian conjecture for 2N variables implies the Dixmier conjecture for N dimension. A self-contained and purely algebraic proof of the last implication is also given by P. K. Adjamagbo and A. van den Essen (2007) who also proved in the same paper that these two conjectures is equivalent to Poisson conjecture.

References

  1. ↑ Tsuchimoto, Yoshifumi (2005). "Endomorphisms of Weyl algebra and $p$-curvatures". Osaka Journal of Mathematics. 42 (2): 435–452. ISSN 0030-6126.
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