Formulation of The Conjecture
The condition JF ≠ 0 is related to the inverse function theorem in multivariable calculus. In fact for smooth functions (and so in particular for polynomials) a local inverse function to F exists at any point where JF is non-zero. However k is algebraically closed so JF as a polynomial will be zero for some complex values of X1, …, XN unless it is a non-zero constant function. Therefore it is a relatively elementary fact that:
Proposition: If F has an inverse function G: kN → kN, then JF is a non-zero constant.
The conjecture is the following strengthening of the converse:
Jacobian conjecture: If JF is a non-zero constant, then F has an inverse function G: kN → kN, and G is regular (in the sense that its components are given by polynomial expressions).
Read more about this topic: Jacobian Conjecture
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