The tree formulas for reversion of power series
Let \(F=(F_ 1,...,F_ n)\) be a system of n power series in n indeterminates \(X_ 1,...,X_ n\) over a field of characteristic 0. Furthermore let \(F_ i\) be of the form \(F_ i=X_ i-H_ i\), \(H_ i\) consisting of terms of degree \(\geq 2\) only. In improving a result of \textit{H. Bass, E. H. Connell} and the author [see Bull. Am. Math. Soc., New Ser. 7, 287-330 (1982; Zbl 0539.13012)] it is shown here without further assumptions that for the analytic inverse \(F^{-1}=G=(G_ 1,...,G_ n)\) with \(X_ i=G_ i(F_ 1,...,F_ n)\) there can be given a summation formula, indexed by certain trees, whose summands are polynomials. Furthermore two related tree formulas are derived, where in one of them (due to J. Towber) every coefficient in G is an expression involving the coefficients of F. Lastly some fundamental relationships between three types of classical polynomials, occurring in this context, are derived, which may be useful in solving the Jacobian conjecture.
scientific article; zbMATH DE number 4099453