Constructive implicit function theorems (Q910438)

Let \(Y=\sum^{\infty}_{n=1}a_ nX^ n \), \(a_ 1\neq 0\), be a formal power series over a commutative ring. In the first part of this paper the author presents an explicit algorithm for the coefficients \(A_ n\) of the reversed series \(X=\sum^{\infty}_{n=1}A_ nY^ n \) in terms of the \(a_ n\). He gives an explicit recurrence relation using partitions of 2n-2 with exactly n-1 parts. In the second part the author proves a constructive implicit function theorem for power series in two variables. This proof also uses direct combinatorial methods. Finally some open problems related to the previous results are listed.