On four-sheeted polynomial mappings of $\mathbb C^2$. I. The case of an irreducible ramification curve

The paper is devoted to the Jacobian Conjecture: a polynomial mapping $f\colon\mathbb C^2\to\mathbb C^2$ with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.