Proper holomorphic mappings between rigid polynomial domains in $\mathbb{C}^{n+1}$

We describe the branch locus of proper holomorphic mappings between rigid polynomial domains in $\mathbb{C}^{n+1}$. It appears, in particular, that it is controlled only by the first domain. As an application, we prove that proper holomorphic self-mappings between such domains are biholomorphic.