Counterexamples to the “Jacobian Conjecture at Infinity”

Earlier, the author constructed an example of an open complex surface $U$, a smooth compact rational curve $L\subset U$ with the self-intersection index $+1$, and a holomorphic immersion $f:U\setminus L\to\mathbb C^2$ that is meromorphic on $U$ but is not an embedding (if $U\subset \mathbb C\mathrm P^2$, then such an immersion can be extended to a counterexample to the Jacobian conjecture). In this paper, an analogous example is constructed with the property that $f|_{\partial U}$ is an immersion of a 3-sphere in $\mathbb C^2$ which is regularly homotopic to an embedding. The map $f$ cannot be extended to a counterexample to the Jacobian conjecture, which is proved by the analysis of the coefficients of polynomials.