I've just been reading the Wikipedia entry regarding the Jacobian conjecture, and it said that either the conjecture is true for all fields of characteristic zero, or it is false for all such fields.
Hence, I wonder, shouldn't this be an easy problem that yields to methods from real or complex analysis? After all, it involves only simple terms like determinant, inverse, constant, polynomial etc.
Specifically, the determinant condition gives a relation between the derivatives, which one may then be able to integrate in order to possibly obtain polynomials.
To make this more specific, say that we have a polynomial function $f: \mathbb K^n \to \mathbb K^n$, where $\mathbb K = \mathbb R$ or $\mathbb C$. Then $\det J_f$ is a polynomial in the derivatives of the components and hence itself a polynomial. By the inverse rule and Cramer's rule, the derivative of the (local) inverse has the form $$ \frac{1}{\det(J_f)} \operatorname{Cof}(J_f), $$ where by assumption $\det(J_f)$ is constant. Also, the cofactor matrix is a polynomial matrix. Thus, we integrate any of its entries for each component to obtain a local polynomial inverse, which is also global due to the identity theorem (at least in the complex case).
What makes this approach fail?
(This main part of my question makes it unique among other questions regarding the Jacobian conjecture, which have been completely falsely suggested to be a duplicate of this one.)
I now see what my mistake was. Instead of being a polynomial in $y$, the variable of the target space, the inverse is a polynomial in $f^{-1}(y)$. Specifically:
$$ J_{f^{-1}}(y) = J_f^{-1}(f^{-1}(y)). $$
Thus, we only obtain a polynomial in the components of $f^{-1}(y)$, which is probably worthless.
The only property of the inverse we have thus shown is this: If we differentiate the function in any direction, we obtain a polynomial in the components of that function. Yet this is even true for $\exp$.