Very degenerate polynomial submersions and counterexamples to the real Jacobian conjecture

The Pinchuk example revisited

“…The main idea is to modify an existing non-injective polynomial map with non-vanishing Jacobian by composing with special mappings. Here we use the version of Pinchuk map presented in [2], because it seems to be the simplest known so far, in the sense that its components have the lowest degree (however this construction still can be adapted to any of the known examples, see [5,7]). Letting p = 4 x 6 y 3 + 12 x 5 y 2 + 12 x 4 y + 4 x 3 y 2 + 4 x 3 + 5 x 2 y + x + y, q = − 4 x 4 y 2 + 8 x 3 y + 4 x 2 + 2 x y + 1…”