The tree formulas for reversion of power series
type: Item
instance of: scholarly article
title: The tree formulas for reversion of power series
zbMATH Open document ID: 0672.13010
DOI: 10.1016/0022-4049(89)90116-3
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13J05
zbMATH DE Number: 4099453
zbMATH Keywords: inverse of system of power series
zbMATH Keywords: analytic inverse
zbMATH Keywords: trees
zbMATH Keywords: Jacobian conjecture
author: David Wright
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90116-3
OpenAlex ID: W1999997928
cites work: Ramification Theoretic Methods in Algebraic Geometry (AM-43)
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Formal inverse expansion and the Jacobian conjecture
description: scientific article; zbMATH DE number 4099453