A generalization of the inversion formulas of systems of power series in systems of implicit functions

Abstract

Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsxj=ϕj(w)=ϕj(w1,...,wm), j=1,...,n, defined by the system of equations fj(w, x)=Fj (w1,...,wm:z1,...,xn)=0, j=1,...,n,fj, (0, 0)=0, ∂Fj(0, 0)/∂zk=δjk in a neighborhood of the point (0, 0)∈C(w,x)m+n, in terms of the coefficients of the power series of the functions Fj(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.