It states that if a polynomial function from an N-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse.

All you have to do is find a function that does not have an inverse, and has a non-zero Det(). The way used by the paper was to find a function which was not one-to-one, while haveing a constant determinant.

Pretty much any math or physics undergrad should be able to work the counter example by the end of their second year.