On counterexamples to Keller's problem (Q1922059)

Let \(F : K^n \to K^n\) be a polynomial map where \(K = \mathbb{R}\) or \(\mathbb{C}\). Let us denote by \(J(F)\) the determinant of the Jacobian of \(F\). The Jacobian conjecture is the following statement: If \(J(F)\) never vanishes then the map \(F\) is injective. Originally, the conjecture was stated for \(K = \mathbb{C}\) with polynomials over \(\mathbb{Z}\) by \textit{O. Keller}. The conjecture for \(K = \mathbb{C}\) is still open for \(n \geq 2\). The conjecture for \(K = \mathbb{R}\), the so-called real Jacobian conjecture was recently shown to be false by \textit{S. Pinchuk}. The main purpose of this paper is to give a proof to the fact that there is no counterexample to the complex Jacobian conjecture of the type constructed by S. Pinchuk for the real case. It is explained how one can view Pinchuk's construction in terms of the asymptotic values of the map. As a consequence it is proved that there are special kinds of polynomial rings such that finding a Jacobian pair within them establishes a counterexample to the conjecture. The simplest such a ring is \(K[V, VU, VU^2 + U]\). For \(K=\mathbb{R}\), Pinchuk found a pair \(P, Q\in\mathbb{R}[V, VU, VU^2 + U]\) whose Jacobian is always positive. We prove that there is no Jacobian pair in \(\mathbb{C}[V, VU, VU^2 + U]\).