Comment by sfpotter
1 day ago
No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.
It's not immediately intuitive what it means for something to be globally and locally invertible. After all, it is obvious that it is both in the 1D case.
You can get the inverse of the Jacobian at any point, but you cannot describe the inverse of the Jacobian through a polynomial, which is a function. You need a more complex object to describe the inverse, because the global inverse is not a function due to the potential of overlapping values.
The determinant of a polynomial mapping is a polynomial, which is the subject of the conjecture. To get the Jacobian determinant, all you need to do is compute partial derivatives of polynomials, and add, subtract, and multiply them together. All of these operations map polynomials to polynomials.
The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.