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
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: kN → kN by setting:
- F(c1, ..., cN) = (f1(c1, ...,cN),..., fN(c1,...,cN))
then JF is itself a polynomial function of the N variables X1, …, XN.
Formulation of the conjecture
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: kN → kN, 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: kN → kN, 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.
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).
- Adjamagbo, P.K.; van den Essen, A. (2007), "A proof of the equivalence of the Dixmier, Jacobian and Poisson conjectures", Acta Math. Vietnam. 32: 205–214
- 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 663785
- 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, MR 2337879
- 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: 140–212, ISSN 0075-4102, MR 691964 Preprint titled "On the global Jacobian conjecture for polynomials of degree less than 100"
- Pinchuk, Sergey (1994), "A counterexample to the strong real Jacobian conjecture", Math. Z. 217 (1): 1–4, MR 1292168
- van den Essen, Arno (2000), Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics 190, Basel: Birkhäuser Verlag, ISBN 3-7643-6350-9, MR 1790619
- van den Essen, A. (2001), "Jacobian conjecture", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- 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
- Wang, Stuart Sui-Sheng (August 1980), "A Jacobian criterion for separability", Journal of Algebra 65: 453–494, doi:10.1016/0021-8693(80)90233-1