Jacobian conjecture
Keller problem
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).
References
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobian_conjecture&oldid=50102