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).