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

Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsx _j =_j(w)=_j(w₁,...,w_m), j=1,...,n, defined by the system of equations f_j(w, x)=F_j (w₁,...,w_m:z₁,...,x _n )=0, j=1,...,n,f _j , (0, 0)=0, F_j(0, 0)/z_k=_jk 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 F_j(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 47–54, January, 1978.