Editing Jacobian conjecture (section)

==Formulation of the conjecture==
It follows from the multivariable chain rule that if ''F'' has a polynomial inverse function ''G'': ''k<sup>N</sup>'' → ''k<sup>N</sup>'', then ''J<sub>F</sub>'' has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

<blockquote>'''Jacobian conjecture:''' Let ''k'' have [[Characteristic (algebra)|characteristic]] 0. If ''J<sub>F</sub>'' is a non-zero constant, then ''F'' has an inverse function ''G'': ''k<sup>N</sup>'' → ''k<sup>N</sup>'' which is [[regular map (algebraic geometry)|regular]], meaning its components are polynomials.</blockquote>

According to van den Essen,<ref name="vdE1997"/> the problem was first conjectured by Keller in 1939 for the limited case of two variables and integer coefficients.

The obvious analogue of the Jacobian conjecture fails if ''k'' has characteristic ''p''&nbsp;>&nbsp;0 even for one variable. The characteristic of a field, if it is not zero, must be prime, so at least 2. The polynomial {{math|''x'' − ''x''<sup>''p''</sup>}} has derivative {{math|1 − ''p x''<sup>''p''−1</sup>}} which is 1 (because ''px'' is 0) but it has no inverse function. However, {{ill|Kossivi Adjamagbo|ht|Pascal Kossivi Adjamagbo}} suggested extending the Jacobian conjecture to characteristic {{nowrap|''p'' > 0}} by adding the hypothesis that ''p'' does not divide the [[Degree of a field extension|degree]] of the [[field extension]] {{nowrap|''k''(''X'') / ''k''(''F'')}}.<ref>{{citation
| mr=1352692
| last=Adjamagbo | first=Kossivi
| chapter=On separable algebras over a U.F.D. and the Jacobian conjecture in any characteristic
| title= Automorphisms of affine spaces (Curaçao, 1994)
| pages= 89–103
| publisher= Kluwer Acad. Publ.
| place= Dordrecht
| year= 1995
| doi=10.1007/978-94-015-8555-2_5 | isbn=978-90-481-4566-9 | doi-access=free}}</ref>

The existence of a polynomial inverse is obvious if ''F'' is simply a set of functions linear in the variables, because then the inverse will also be a set of linear functions. A simple non-linear example is given by

:<math>u=x^2+y+x</math>
:<math>v=x^2+y</math>

so that the Jacobian determinant is

:<math>J_F = \left | \begin{matrix} 1+2x & 1 \\
2x & 1 \end{matrix} \right |
= (1+2x)(1) - (1)2x = 1.
</math>

In this case the inverse exists as the polynomials

:<math>x=u-v</math>
:<math>y=v-(u-v)^2.</math>

But if we modify ''F'' slightly, to

:<math>u=2x^2+y</math>
:<math>v=x^2+y</math>

then the determinant is

:<math>J_F = \left | \begin{matrix} 4x & 1 \\
2x & 1 \end{matrix} \right |
= (4x)(1) - 2x(1) = 2x,
</math>

which is not constant, and the Jacobian conjecture does not apply.
The function still has an inverse:

:<math>x=\sqrt{u-v}</math>
:<math>y=2v-u,</math>

but the expression for ''x'' is not a polynomial.

The condition ''J<sub>F</sub>'' ≠ 0 is related to the [[inverse function theorem]] in [[multivariable calculus]]. In fact for smooth functions (and so in particular for polynomials) a smooth local inverse function to ''F'' exists at every point where ''J<sub>F</sub>'' is non-zero. For example, the map x → ''x''&nbsp;+&nbsp;''x''<sup>3</sup> has a smooth global inverse, but the inverse is not polynomial.