We describe the branch locus of proper holomorphic mappings between rigid polynomial domains in 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.

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Abstract. The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping f : D → Ω with branch locus V f is factored by automorphisms if and only if f * (π 1 (D\f

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