Pinchuk's non-injective local diffeomorphisms of the plane

Pinchuk's maps are local diffeomorphisms of the plane that are not one-to-one. They refute the strong real Jacobian conjecture, because their Jacobian is positive everywhere and they still fold the plane over itself.^[1] Their Jacobian is not constant, so they are not Keller maps and do not bear on JC₂ directly. What they show is that no proof of JC₂ can rest on real local invertibility and topology alone: any proof of the plane conjecture has to use that the determinant is constant, or that the field is ℂ.

Campbell drew one member of the class. The image misses exactly the two points $(0,0)$ and $(-1,-\tfrac{163}4)$, and its asymptotic variety is a connected curve, parametrized by polynomials of degrees 2 and 5.^[2] That set is where the map fails to be proper, and Pinchuk's map is the standard worked example of a real non-properness set.

1. ↑ S. Pinchuk, A counterexamle [sic] to the strong real Jacobian conjecture, Math. Z. 217 (1994) 1–4, doi:10.1007/BF02571929.
2. ↑ L. A. Campbell, Picturing Pinchuk's Plane Polynomial Pair, arXiv:math/9812032 (1998).