Proposition 9.2.2 (Weierstrass division)611958EF0AB8

Proof. Polydiscs as in the statement exist with arbitrarily small radii, because the roots of $P(z',\cdot )$ tend to $0$ as $z'\to 0$, by (9.2.1).

Let $\Delta '\times \Delta $ be such a polydisc and $f$ a holomorphic function on it. For $\rho <s<r$ and $|w|<s$ define $q$ and $R$ by (9.2.3). For fixed $z'$ and $w$, both integrands are holomorphic in $\zeta $ on the annulus $\{ \rho <|\zeta |<r\} $, except for the pole of the first one at $\zeta =w$; by Cauchy's theorem the integrals do not depend on $s$ as long as $|w|<s$, and the second one does not depend on $s\in (\rho ,r)$ at all. Hence $q$ and $R$ are defined on $\Delta '\times \Delta $, and they are holomorphic by differentiation under the integral sign. The difference quotient $(P(z',\zeta )-P(z',w))/(\zeta -w)$ is a polynomial in $w$ of degree $m-1$ whose coefficients are polynomials in $\zeta $ and in the $a_ j(z')$; hence $R$ is a polynomial in $w$ of degree less than $m$ with holomorphic coefficients on $\Delta '$. Adding the two integrals, we obtain $Pq+R=(2\pi \mathrm i)^{-1}\int _{|\zeta |=s}f(z',\zeta )(\zeta -w)^{-1}\, \mathrm d\zeta =f$ by Cauchy's formula.

For the estimate, let $f$ be bounded, and fix $s_0=(\rho +r)/2$. For $|\zeta |\geq s_0$ we have $|P(z',\zeta )|=\prod _ l|\zeta -\lambda _ l(z')|\geq (s_0-\rho )^ m$, where $\lambda _ l(z')$ are the roots, and the coefficients of the difference quotient are bounded for $z'\in \Delta '$, $|\zeta |=s_0$ and $|w|<r$, since the $a_ j$ are bounded on $\Delta '$. The integral over $|\zeta |=s_0$ therefore gives $|R|\leq C_1\sup |f|$ on $\Delta '\times \Delta $. For $s_0\leq |w|<r$ we have $|q|=|f-R|/|P|\leq (1+C_1)(s_0-\rho )^{-m}\sup |f|$, and the maximum principle for the holomorphic functions $q(z',\cdot )$ on the discs $\{ |w|\leq s\} $, $s_0\leq s<r$, extends this bound to all of $\Delta '\times \Delta $.

For uniqueness, suppose that $Pq+R=0$ on a polydisc as above, with $R$ a polynomial in $w$ of degree less than $m$. For each $z'$, the polynomial $R(z',\cdot )=-P(z',\cdot )q(z',\cdot )$ vanishes at the $m$ roots of $P(z',\cdot )$, counted with multiplicity, all of which lie in $\Delta $. Its degree is less than $m$, so $R(z',\cdot )=0$. Then $P(z',\cdot )q(z',\cdot )=0$, and since $P(z',\cdot )$ has only finitely many zeros, the continuous function $q(z',\cdot )$ vanishes on a dense set, hence everywhere. Finally, every germ is represented by a bounded holomorphic function on some polydisc as above, and an identity between germs holds on some such polydisc; so the assertions for germs follow. $\square$