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 functions $z_j=\varphi_j(w)=\varphi_j(w_1,\dots,w_m)$, $j=1,\dots,n$, defined by the system of equations $F_j(w,z)=F_j(w_1,\dots,2_m;z_1,\dots,z_n)=0$, $j=1,\dots,n$, $F_j(0,0)=0$, $\partial F_j(0,0)/\partial z_k=\delta_{jk}$ in a neighborhood of the point $(0,0)\in C_{(w,z)}^{m+n}$, in terms of the coefficients of the power series of the functions $F_j(w,z)$, $j=1,\dots,n$. As a corollary, well-known formulas are obtained for the inversion of multiple power series.