TITLE: Example of nonlinear regular function with constant nonzero Jacobian
QUESTION [5 upvotes]: Can anyone give a nonlinear regular function from C^2 to C^2 with a constant nonzero Jacobian? It seems to me that the only such functions are linear. 
According to the Jacobian conjecture, a function from C^2 to C^2 with a constant nonzero Jacobian must have an inverse.

REPLY [3 votes]: Any linear function has constant Jacobian determinant, as does any map of the form $(z_1,z_2) \rightarrow (z_1, z_2 - f(z_1))$ or $(z_1,z_2) \rightarrow (z_1 - f(z_2), z_2)$. As a result, any finite composition of maps of these forms will have constant Jacobian. These include the examples that Shai Covo and Andrew Marshall listed. I forget if there are known examples outside this category.