On finite regular and holomorphic mappings

We show that for every hypersurface V⊂Y and every k∈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 μ(f)=k. In particular if Knr={x∈Cn:∏i=1rxi=0} and X is a smooth and simply connected algebraic manifold, then every finite regular mapping f:X→Cn with D(f)=Knr is equivalent to one of the mappings fd1,…,dr:Cn∋(x1,…,xn)↦(x1d1,…,xrdr,xr+1,…,xn)∈Cn. 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:(Cn,0)→(Cn,0) is a proper and holomorphic mapping of topological degree two, then there exist biholomorphisms Ψ,Φ:(Cn,0)→(Cn,0) such that Ψ∘f∘Φ(x1,x2,…,xn)=(x12,x2,…,xn). Moreover, for every proper holomorphic mapping f:(Cn,0)→(Cn,0) which has a discriminant with only simple normal crossings, there exist biholomorphisms Ψ,Φ:(Cn,0)→(Cn,0) such that Ψ∘f∘Φ(x1,x2,…,xn)=(x1d1,x2d2,…,xrdr,xr+1,…,xn), where r is the number of irreducible components of the discriminant at 0.