Editing Jacobian conjecture (section)

==The Jacobian determinant==
Let ''N'' > 1 be a fixed integer and consider  polynomials ''f''<sub>1</sub>, ..., ''f''<sub>''N''</sub> in variables ''X''<sub>1</sub>, ..., ''X''<sub>''N''</sub> with [[coefficient]]s  in a field ''k''. Then we define a [[vector-valued function]] ''F'': ''k<sup>N</sup>'' → ''k''<sup>''N''</sup> by setting:

: ''F''(''X''<sub>1</sub>, ..., ''X''<sub>''N''</sub>) = (''f''<sub>1</sub>(''X''<sub>1</sub>, ...,''X''<sub>''N''</sub>),..., ''f''<sub>''N''</sub>(''X''<sub>1</sub>,...,''X''<sub>''N''</sub>)).

Any map ''F'': ''k<sup>N</sup>'' → ''k''<sup>''N''</sup> arising in this way is called a [[polynomial mapping]].

The [[Jacobian matrix and determinant|Jacobian determinant]] of ''F'', denoted by ''J<sub>F</sub>'', is defined as the [[determinant]] of the ''N'' × ''N'' [[Jacobian matrix]] consisting of the [[partial derivative]]s of ''f<sub>i</sub>'' with respect to ''X<sub>j</sub>'':

:<math>J_F = \left | \begin{matrix} \frac{\partial f_1}{\partial X_1} & \cdots & \frac{\partial f_1}{\partial X_N} \\
\vdots & \ddots & \vdots \\
\frac{\partial f_N}{\partial X_1} & \cdots & \frac{\partial f_N}{\partial X_N} \end{matrix} \right |,</math>

then ''J<sub>F</sub>'' is itself a polynomial function of the ''N'' variables ''X''<sub>1</sub>, ..., ''X<sub>N</sub>''.