This exercise is from Greuel & Lossen & Shustin's Introduction to Singularities and Deformations, Exercise 1.2.5.
Let $f\in\mathbf{C}\langle \mathbf{x},y\rangle$, where $\mathbf{C}\langle \mathbf{x},y\rangle$ is the ring of convergent power series with $n+1$ variables, namely $\mathbf{x}=(x_1,\ldots,x_n)$ and $y$. If $f\in \langle y\rangle +\langle \mathbf{x}\rangle^k\subset\mathbf{C}\langle \mathbf{x},y\rangle$ and $\frac{\partial f}{\partial y}(\mathbf{0})\neq 0,$ i.e. the coefficient of $y$ is nonzero, then prove that there exists $Y\in \langle \mathbf{x}\rangle^k\subset\mathbf{C}\langle \mathbf{x}\rangle$ such that $f(\mathbf{x},Y)=0$.
My attempt: If $k=1$, then it is merely the direct consequence of implicit function theorem. However, the problem requires to prove much stronger result. First I attempted as follows:
Let $f=yg+h$ where $g\in \mathbf{C}\langle \mathbf{x},y\rangle$ and $h\in \langle \mathbf{x}\rangle^k.$ Then I simply let $Y=-g^{-1}h$, but I found that $Y$ is not necessarily in $\mathbf{C}\langle \mathbf{x}\rangle$. How should I find $Y$ such that $Y\in \mathbf{C}\langle \mathbf{x}\rangle$ satisfying $f(\mathbf{x},Y)=0$?
Thanks in advance!
Let $f(\mathbf{x},y)=yg(\mathbf{x},y)+h(\mathbf{x},y),$ where $g\in \mathbf{C}\langle\mathbf{x},y\rangle$ and $h\in \langle\mathbf{x}\rangle^k$. Then $f(\mathbf{0},0)=0$ and by assumption $\frac{\partial f}{\partial y}(\mathbf{0},0)\neq 0.$ Therefore, by implicit function theorem, there exists a $Y(\mathbf{x})\in \langle\mathbf{x}\rangle\subset\mathbf{C}\langle \mathbf{x}\rangle$ such that $f(\mathbf{x},Y(\mathbf{x}))=0.$ This means $Y(\mathbf{x})g(\mathbf{x},Y(\mathbf{x}))+h(\mathbf{x},Y(\mathbf{x}))=0.$ Note that $g(\mathbf{x},Y(\mathbf{x}))$ is a unit and $h(\mathbf{x},Y(\mathbf{x}))\in \langle x\rangle^k.$ Then it must be the case that $Y(\mathbf{x})\in \langle \mathbf{x}\rangle^k$.