Structure of Keller mappings, two-dimensional case

A Keller map is a polynomial mapping $f: \Bbb R^n \to \Bbb R^n$ (or $\Bbb C^n \to \Bbb C^n$) with the Jacobian $J_f\equiv \mathrm{const}\ne0$. The Jacobian conjecture was first formulated by O. N. Keller in 1939. In the modern form it supposes injectivity of a Keller map. Earlier, in the case $n=2$, the author gave a complete description of Keller maps with $\deg f\le 3.$ This paper is devoted to the description of Keller maps for which $\deg f\le 4.$ Significant differences between these two cases are revealed, which, in particular, indicate the irregular structure of Keller maps even in the case of $n=2$.