Exercise 4, § 2

Let $n$ be an integer $\geqslant 1$ and let U be a connected open neighbourhood of 0 in $\mathbf{C}^n$.

Let $f: U\rightarrow \mathbf{C}$ be a nonconstant analytic mapping. Put $A =\overset{-1}{f}(0)$; suppose that A is not empty.

(a) Prove that there exist a point $a\in$ A and an integer $p\geqslant 1$ such that $D^mf(z) = 0$ for all $z\in A$ and all $m\in \{0, . . . , p\}$ and $D^{p+1}f(a)= 0\not$ .

(b) Let $(\alpha_1, . . . , \alpha_n)$ be a family of positive integers and let $m\in \{1, . . . , n\}$ such that $\alpha_1+\cdots +\alpha_n=p$ and $\partial_m\partial^{\alpha}f(a)= 0\not$ . Let B be the set of points $z\in U$ such that $\partial^{\alpha}f(z) = 0$. Prove that there exists a neighbourhood V of $a$ contained in U such that $B\cap V$ is an analytic submanifold of V of dimension $(n-1)$. (c) Prove that there exists a neighbourhood W of $a$ contained in V such that $A\cap W = B\cap W$. (Reduce to the case where $m=n$ and put $a= (a', a_n)$; prove that for every point $w'\in \mathbf{C}^{n-1}$ sufficiently close to $a'$, there exists a point $w\in \mathbf{C}$ such that $(w', w)\in V$ and $f(w', w) = 0$.)

(d) Prove that there exists a closed subset S of A, with empty interior in A, such that A-S is an analytic submanifold of dimension $(n-1)$.