Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous unsolved problem on 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.

Jacobian conjecture

Field	Algebraic geometry
Conjectured by	Ott-Heinrich Keller
Conjectured in	1939
Equivalent to	Dixmier conjecture

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 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 was numbered 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:

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

then 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 (1997), 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 must be prime, so it is 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, 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).

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

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, ..., X_n + H_n), where each H_i 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 H_i 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, ..., 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 Campbell (1973) for complex maps and in general by Razar (1979) and, independently, Wright (1981). Moh (1983) checked the conjecture for polynomials of degree at most 100 in two variables.

de Bondt, van den Essen & 2005, 2005 harvtxt error: no target: CITEREFde_Bondtvan_den_Essen2005,_2005 (help) 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), 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

• Adjamagbo, Kossivi (1995), "On separable algebras over a U.F.D. and the Jacobian conjecture in any characteristic", Automorphisms of affine spaces (Curaçao, 1994), Dordrecht: Kluwer Acad. Publ., pp. 89–103, MR 1352692
• Adjamagbo, P. K.; van den Essen, A. (2007), "A proof of the equivalence of the Dixmier, Jacobian and Poisson conjectures" (PDF), Acta Math. Vietnam., 32: 205–214, MR 2368008
• Bass, Hyman; Connell, Edwin H.; Wright, David (1982), "The Jacobian conjecture: reduction of degree and formal expansion of the inverse", American Mathematical Society. Bulletin. New Series, 7 (2): 287–330, doi:10.1090/S0273-0979-1982-15032-7, ISSN 1088-9485, MR 0663785
• Belov-Kanel, Alexei; Kontsevich, Maxim (2007), "The Jacobian conjecture is stably equivalent to the Dixmier conjecture", Moscow Mathematical Journal, 7 (2): 209–218, arXiv:math/0512171, Bibcode:2005math.....12171B, doi:10.17323/1609-4514-2007-7-2-209-218, MR 2337879
• Campbell, L. Andrew (1973), "A condition for a polynomial map to be invertible", Math. Ann., 205 (3): 243–248, doi:10.1007/bf01349234, MR 0324062 (48 #2414)
• Connell, E.; van den Dries, L. (1983), "Injective polynomial maps and the Jacobian conjecture", J. Pure Appl. Algebra, 28 (3): 235–239, doi:10.1016/0022-4049(83)90094-4, MR 0701351
• de Bondt, Michiel; van den Essen, Arno (2005), "A reduction of the Jacobian conjecture to the symmetric case", Proc. Amer. Math. Soc., 133 (8): 2201–2205 (electronic), doi:10.1090/S0002-9939-05-07570-2, MR 2138860
• de Bondt, Michiel; van den Essen, Arno (2005), "The Jacobian conjecture for symmetric Drużkowski mappings", Ann. Polon. Math., 86 (1): 43–46, doi:10.4064/ap86-1-5, MR 2183036
• Drużkowski, Ludwik M. (1983), "An effective approach to Keller's Jacobian conjecture", Math. Ann., 264 (3): 303–313, doi:10.1007/bf01459126, MR 0714105
• Drużkowski, Ludwik M. (2005), "The Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case", Ann. Polon. Math., 87: 83–92, doi:10.4064/ap87-0-7, MR 2208537
• Keller, Ott-Heinrich (1939), "Ganze Cremona-Transformationen", Monatshefte für Mathematik und Physik, 47 (1): 299–306, doi:10.1007/BF01695502, ISSN 0026-9255
• Moh, T. T. (1983), "On the Jacobian conjecture and the configurations of roots", Journal für die reine und angewandte Mathematik, 340 (340): 140–212, doi:10.1515/crll.1983.340.140, ISSN 0075-4102, MR 0691964
• Moh, T. T., On the global Jacobian conjecture for polynomials of degree less than 100, preprint
• Pinchuk, Sergey (1994), "A counterexample to the strong real Jacobian conjecture", Math. Z., 217 (1): 1–4, doi:10.1007/bf02571929, MR 1292168
• Razar, Michael (1979). "Polynomial maps with constant Jacobian". Israel J. Math. 32 (2–3): 97–106. doi:10.1007/bf02764906. MR 0531253. (80m:14009)
• van den Essen, Arno (2000), Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Basel: Birkhäuser Verlag, doi:10.1007/978-3-0348-8440-2, ISBN 978-3-7643-6350-5, MR 1790619
• van den Essen, Arno (1997), "Polynomial automorphisms and the Jacobian conjecture" (PDF), Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), Sémin. Congr., 2, Paris: Soc. Math. France, pp. 55–81, MR 1601194
• Tsuchimoto, Yoshifumi (2005). "Endomorphisms of Weyl algebra and $p$-curvatures". Osaka Journal of Mathematics. 42 (2): 435–452. ISSN 0030-6126.
• Wang, Stuart Sui-Sheng (August 1980), "A Jacobian criterion for separability", Journal of Algebra, 65 (2): 453–494, doi:10.1016/0021-8693(80)90233-1
• Wright, David (1981). "On the Jacobian conjecture". Illinois J. Math. 25 (3): 423–440. MR 0620428. (83a:12032)

External links

• Web page of T. T. Moh on the conjecture