On finite regular and holomorphic mappings

We show that for every hypersurface V subset of Y and every k is an element of N, there are only a finite number of non-equivalent finite regular mappings f : X -> Y such that the discriminant D(f) equals V and mu(f) = k. In particular if K-n(r) = {x is an element of C-n : Pi(r)(i=1) x(i) = 0} and X is a smooth and simply connected algebraic manifold, then every finite regular mapping f : X -> C-n with D(f) = K-n(r) is equivalent to one of the mappings f(d1,) ...(,) (dr) : C-n (sic) (x(1,) ...(,) x(n)) ->(x(1)(d1), ..., x(r)(dr), x(r+1), ..., x(n)) is an element of C-n. Moreover, we obtain generalizations of the Lamy Theorem. We prove the same statement in the local (and sometimes global) holomorphic situation. In particular we show that if f : (C-n, 0) -> (C-n, 0) is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms Psi, Phi : (C-n, 0) -> (C-n, 0) such that Psi . f . Phi(x(1), x(2), ..., x(n)) = (x(1)(2), x(2), ..., x(n)). Moreover, for every proper holomorphic mapping f : (C-n, 0) -> (C-n, 0) which has a discriminant with only simple normal crossings, there exist biholomorphisms Psi,Phi : (C-n, 0) -> (C-n, 0) such that Psi . f . Phi (x(1), x(2), ..., x(n)) = (x(1)(d1), x(2)(d2), ..., x(r)(dr), x(r+1), ..., x(n)), where r is the number of irreducible components of the discriminant at 0. (C) 2016 Elsevier Inc. All rights reserved.