Theorem 5.5.2 (Weierstrass preparation).F54A645FF399 Let $f$ be holomorphic on a neighborhood $U$ of $0$ in $\mathbb {C}^{n-1}\times \mathbb {C}$, $n\geq 1$, and suppose that $w\mapsto f(0,w)$ has a zero of order $m\ge 1$ at $w=0$. Then there are polydiscs $P'\times \{ |w|<\rho \} \subset U$ centered at $0$ with arbitrarily small radii, and numbers $\rho _1\in (0,\rho )$, such that on $P'\times \{ |w|<\rho \} $
\[ f(z',w)=g(z',w)W(z',w),\qquad W=w^ m+a_1(z')w^{m-1}+\cdots +a_ m(z'), \]
where $g$ is holomorphic and nowhere zero, the coefficients $a_ j$ are holomorphic on $P'$ with $a_ j(0)=0$, and for every $z'\in P'$ the $m$ roots of $W(z',\cdot )$ lie in $\{ |w|<\rho _1\} $.
Proof.
Since the zeros of $f(0,\cdot )$ are isolated, there is $\rho _0>0$ such that $\{ 0\} \times \{ |w|\leq \rho _0\} \subset U$ and $f(0,w)\neq 0$ for $0<|w|\leq \rho _0$. Let $0<\rho <\rho _0$, and choose $\rho _1<\rho <\rho _2\leq \rho _0$. Then $f(0,w)\neq 0$ on the compact annulus $\{ \rho _1\leq |w|\leq \rho _2\} $. Since $|f(0,\cdot )|$ has a positive minimum on this annulus and $f$ is uniformly continuous on a compact neighborhood of $\{ 0\} \times \{ \rho _1\leq |w|\leq \rho _2\} $ in $U$, there is a polydisc $P'$ centered at $0$, as small as we wish, such that $P'\times \{ |w|\leq \rho _2\} \subset U$ and $f(z',w)\neq 0$ for $z'\in P'$ and $\rho _1\leq |w|\leq \rho _2$. By the argument principle, the number of zeros of $f(z',\cdot )$ in $\{ |w|<\rho \} $, counted with multiplicity, is
\[ \frac{1}{2\pi \mathrm{i}}\int _{|w|=\rho }\frac{f_ w(z',w)}{f(z',w)}\, \mathrm{d}w, \]
a continuous integer-valued function of $z'$ on the connected set $P'$; hence it equals its value $m$ at $z'=0$. Denote these zeros by $\lambda _1(z'),\ldots ,\lambda _ m(z')$, in any order; they lie in $\{ |w|<\rho _1\} $.
By the residue theorem, their power sums are
\[ p_ k(z')=\sum _{l=1}^ m\lambda _ l(z')^ k=\frac{1}{2\pi \mathrm{i}}\int _{|w|=\rho } w^ k\frac{f_ w(z',w)}{f(z',w)}\, \mathrm{d}w,\qquad 1\leq k\leq m, \]
which are holomorphic functions on $P'$, by differentiation under the integral sign. The elementary symmetric functions $e_ k$ of the roots satisfy Newton's identities
\[ ke_ k=\sum _{j=1}^ k(-1)^{j-1}e_{k-j}p_ j,\qquad e_0=1. \]
To prove them, put $E(t)=\prod _ l(1+t\lambda _ l)=\sum _ ke_ kt^ k$ and compare the coefficients of $t^{k-1}$ in $E'(t)=E(t)\sum _{j\geq 1}(-1)^{j-1}p_ jt^{j-1}$, which is the expansion of the logarithmic derivative $E'/E=\sum _ l\lambda _ l/(1+t\lambda _ l)$. By induction on $k$, every $e_ k$ is holomorphic on $P'$. Hence $W(z',w)=\prod _ l(w-\lambda _ l(z'))$ is a monic polynomial in $w$ with holomorphic coefficients $a_ k=(-1)^ ke_ k$, and $a_ k(0)=0$ because all $\lambda _ l(0)=0$.
It remains to show that $f/W$ extends to a nowhere vanishing holomorphic function on $P'\times \{ |w|<\rho \} $. For fixed $z'\in P'$, the functions $f(z',\cdot )$ and $W(z',\cdot )$ have the same zeros, with the same multiplicities, in $\{ |w|<\rho _2\} $; so $f/W$ and $W/f$ extend to holomorphic functions of $w$ there. On the annulus $\{ \rho _1<|w|<\rho _2\} $ both quotients are holomorphic in $(z',w)$. The Cauchy integral
\[ G(z',w)=\frac{1}{2\pi \mathrm i}\int _{|\zeta |=\rho }\frac{(f/W)(z',\zeta )}{\zeta -w}\, \mathrm d\zeta ,\qquad |w|<\rho , \]
is holomorphic in $(z',w)$, by differentiation under the integral sign, and equals $f/W$ by Cauchy's formula in $w$. Defining $\widetilde G$ in the same way from $W/f$, we obtain $G\widetilde G=1$. Thus $g=G$ is holomorphic and nowhere zero, and $f=gW$.
$\square$
Pome
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