Let be a polynomial mapping, i.e. each is a polynomial in variables. If has a polynomial mapping as an inverse, then the chain rule implies that the determinant of the Jacobi matrix is a non-zero constant. In 1939, O.H. Keller asked: is the converse true?, i.e. does imply that has a polynomial inverse?, [a4]. This problem is now known as Keller's problem but is more often called the Jacobian conjecture. This conjecture is still open (1999) for all . Polynomial mappings satisfying are called Keller mappings. Various special cases have been proved:
1) if , the conjecture holds (S.S. Wang). Furthermore, it suffices to prove the conjecture for all and all Keller mappings of the form where each is either zero or homogeneous of degree (H. Bass, E. Connell, D. Wright, A. Yagzhev). This case is referred to as the cubic homogeneous case. In fact, it even suffices to prove the conjecture for so-called cubic-linear mappings, i.e. cubic homogeneous mappings such that each is of the form , where each is a linear form (L. Drużkowski). The cubic homogeneous case has been verified for ( was settled by D. Wright; was settled by E. Hubbers).
2) A necessary condition for the Jacobian conjecture to hold for all is that for Keller mappings of the form with all non-zero coefficients in each positive, the mapping is injective (cf. also Injection), where denotes the homogeneous part of degree of . It is known that this condition is also sufficient! (J. Yu). On the other hand, the Jacobian conjecture holds for all and all Keller mappings of the form , where each non-zero coefficient of all is negative (also J. Yu).
3) The Jacobian conjecture has been verified under various additional assumptions. Namely, if has a rational inverse (O.H. Keller) and, more generally, if the field extension is a Galois extension (L.A. Campbell). Also, properness of or, equivalently, if is finite over (cf. also Extension of a field) implies that a Keller mapping is invertible.
4) If , the Jacobian conjecture has been verified for all Keller mappings with (T.T. Moh) and if or is a product of at most two prime numbers (H. Applegate, H. Onishi). Finally, if there exists one line such that is injective, then a Keller mapping is invertible (J. Gwozdziewicz). There are various seemingly unrelated formulations of the Jacobian conjecture. For example,
a) up to a polynomial coordinate change, is the only commutative -basis of ;
b) every order-preserving -endomorphism of the th Weyl algebra is an isomorphism (A. van den Essen).
c) for every there exists a constant such that for every commutative -algebra and every with and , one has (H. Bass).
d) if is a polynomial mapping such that for some , then for some .
e) if, in the last formulation, one replaces by the so-called real Jacobian conjecture is obtained, i.e. if is a polynomial mapping such that for all , then is injective. It was shown in 1994 (S. Pinchuk) that this conjecture is false for . Another conjecture, formulated by L. Markus and H. Yamabe in 1960 is the global asymptotic stability Jacobian conjecture, also called the Markus–Yamabe conjecture. It asserts that if is a -mapping with and such that for all the real parts of all eigenvalues of are , then each solution of tends to zero if tends to infinity. The Markus–Yamabe conjecture (for all ) implies the Jacobian conjecture. For the Markus–Yamabe conjecture was proved to be true (R. Fessler, C. Gutierrez). However, in 1995 polynomial counterexamples where found for all (A. Cima, A. van den Essen, A. Gasull, E. Hubbers, F. Mañosas).
|[a1]||A. van den Essen, "Polynomial automorphisms and the Jacobian conjecture" J. Alev (ed.) et al. (ed.) , Algèbre Noncommutative, Groupes Quantiques et Invariants , SMF (1985) pp. 55–81|
|[a2]||A. van den Essen, "Seven lectures on polynomial automorphisms" A. van den Essen (ed.) , Automorphisms of Affine Spaces , Kluwer Acad. Publ. (1995) pp. 3–39|
|[a3]||H. Bass, E.H. Connell, D. Wright, "The Jacobian conjecture: reduction of degree and formal expansion of the inverse" Bull. Amer. Math. Soc. , 7 (1982) pp. 287–330|
|[a4]||O.H. Keller, "Ganze Cremonatransformationen" Monatschr. Math. Phys. , 47 (1939) pp. 229–306|
|[a5]||A. van den Essen, "Polynomial automorphisms and the Jacobian conjecture" , Birkhäuser (to appear in 2000)|
Jacobian conjecture. A. van den Essen (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Jacobian_conjecture&oldid=19054