Taylor expansion of implicit functions defined by linear equations of variables (Q5940265)

Let \(y\) be the implicit function defined by the equation \(g(y)= \sum^s_{i=1} x_if_i(y)\), where \(f_1=1\) and \(g\), \(f_2,\dots,f_s\) are holomorphic at \(y^0=y(x_1^0,0, \dots,0)\). The author obtains the Taylor expansion of \(y\) at \((x_0,\dots, x_s)=0\), and then applies his method to the general case: \(f(x_1,\dots, x_n;y)=0\), with \(f_y(c_1, \dots,c_n; y^0)\neq 0\).