- Jacobian conjecture
In
mathematics , the Jacobian conjecture is a celebrated problem onpolynomial s in severalvariable s. It was first posed in 1939 byOtt-Heinrich Keller . It was later named and widely publicised byShreeram Abhyankar , as an example of a question in the area ofalgebraic geometry that requires little beyond a knowledge ofcalculus to state.Formulation
For fixed "N" > 1 consider "N" polynomials "F""i", for 1 ≤ "i" ≤ "N" in the variables
:"X"1, …, "X""N",
and with
coefficient s in analgebraically closed field "k" (in fact, it suffices to assume "k"="C", the field of complex numbers). TheJacobian determinant "J" of the "F""i", considered as a vector-valued function:"F": "k""N" → "k""N",
is by definition the
determinant of the "N" × "N" matrix of the:"F""ij",
where "F""ij" is the
partial derivative of "F""i" with respect to "X""j".The condition
:"J" ≠ 0
enters into the
inverse function theorem inmultivariable calculus . In fact that condition for smooth functions (and so a fortiori for polynomials) ensures the existence of a local inverse function to "F", at any point where it holds.On the other hand in the polynomial case "J" is itself a polynomial. Since "k" is algebraically closed, "J" will be zero for some complex values of "X"1, …, "X""N", "unless" we have the condition
:"J" is a
constant .Therefore it is a relatively elementary fact that
:if "F" has an
inverse function defined everywhere, then "J" is a constant.The Jacobian conjecture is the
converse : it states that:if "J" is a non-zero constant function, then "F" has an inverse function.
The Jacobian conjecture has been proved for polynomials of degree 2, and it has also been shown that it follows from the special case where the polynomials are of degree 3.
The Jacobian conjecture is notorious for the large number of attempted proofs that turned out to contain subtle errors. There are currently no plausible claims to have proved it.
It has been proved that the Jacobian conjecture is equivalent to the
Dixmier conjecture .ee also
*
Smale's problems
*Dixmier conjecture References
*springer|id=J/j120010|title=Jacobian conjecture|author=A. van den Essen
*O.H. Keller, "Ganze Cremonatransformationen" Monatschr. Math. Phys. , 47 (1939) pp. 229–306
*A. van den Essen, "Polynomial automorphisms and the Jacobian conjecture", ISBN 3-7643-6350-9External links
* [http://www.math.purdue.edu/~ttm/jacobian.html Web page of T. T. Moh on the conjecture]
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