Roots of formal power series in one variable (Q1066369)

Let \(\Omega\) be the set of formal power series \(f=\sum^{\infty}_{i=1}a_ i x^ i\) in one variable. This set together with the composition \(f\circ g\) of formal power series, forms a semigroup. Let \(g^{(m)}\) denote the m-th iterative power of g, i.e. the m-th power of g with respect to composition. The author deals with the problem of determining for given f all integers \(m_ j\) such that there exists some \(g\in \Omega\) with \(g^{(m)}=f\). The author's main theorem provides several sufficient conditions for \(f\in \Omega\) to have roots of order m.