Jacobian Conjecture - The Jacobian Determinant

The Jacobian Determinant

Let N > 1 be a fixed integer and consider the polynomials f₁, ..., f_N in variables X₁, ..., X_N with coefficients in an algebraically closed field k (in fact, it suffices to assume k = C). Then we define a vector-valued function F: kN → kN by setting:

F(c₁, ..., c_N) = (f₁(c₁), ..., f_N(c_N))

The Jacobian determinant of F, denoted by J_F, is defined as the determinant of the N × N matrix consisting of the partial derivatives of f_i with respect to X_j:

$J_F = \left | \begin{matrix} \frac{\partial f_1}{\partial X_1} & \cdots & \frac{\partial f_1}{\partial X_N} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_N}{\partial X_1} & \cdots & \frac{\partial f_N}{\partial X_N} \end{matrix} \right |,$

then J_F is itself is a polynomial function of the N variables X₁, …, X_N.

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