On the nonexistence of iterative roots

For the germ of a holomorphic mapping F:(U,0)-->(C,0) of the form F(z)=qz+...., where q is a primitive root of unity of order d greater than or equal to 2, criteria for the existence of a continuous iterative root of given order and the topological linearizability of F are given. The following conditions are equivalent: (1) F-d = Id; (2) the germ of the mapping F(z) is topologically conjugate to the germ of the mapping qz; (3) the germ of the mapping F has a continuous iterative root of order d(k) for every k greater than or equal to 1. If F-d not equal Id, then for a given positive integer N the germ of the mapping F has a continuous iterative root of order N iff d . gcd(N,ind(F-d,0)-1) divides ind(F-d,0)-1. (C) 1997 Elsevier Science B.V.