Lemma 6.12.1BCE8B698A66D

Proof. (1) If $h=w_1^ kh_1$ near a point $(0,w'_0)$ of $E\cap \Delta $, with $h_1$ holomorphic, then the Leibniz rule gives $\partial _{w_1}^ lh(0,w')=0$ for $l<k$ and $w'$ near $w'_0$; so $a_ l=0$ near $w'_0$, and hence on the connected set $\Delta '$ by the identity theorem. Conversely, if $a_ l=0$ for $l<k$, then $h=w_1^ k\sum _{l\geq k}a_ lw_1^{l-k}$ near $p$; the last series is obtained from the Taylor series of $h$ at $p$ by lowering the exponent of $w_1$ in each term by $k$, so it converges on the same polydiscs and defines a holomorphic function (section 9.2). If $h$ vanishes on $E$ near $p$, then $a_0=h(0,\cdot )$ vanishes near the center of $\Delta '$, hence on $\Delta '$, and the case $k=1$ shows that $h$ is divisible by $w_1$ near $p$. Applying this to $\tilde w_1$, and to $w_1$ with the roles of the two coordinate systems exchanged, we obtain $\tilde w_1=c\, w_1$ and $w_1=\tilde c\, \tilde w_1$ near $p$ with holomorphic $c$ and $\tilde c$. Then $c\tilde c=1$ off $E$; since the continuous function $c\tilde c$ equals $1$ on the dense set $\{ w_1\neq 0\} $, it equals $1$ near $p$, and $c$ is a unit.

(2) By (1), $h$ is divisible by $w_1^ k$ near $p$ if and only if it is divisible by $\tilde w_1^ k$ near $p$, so the order does not depend on the coordinates. Let $k_0$ be the smallest $l$ with $a_ l\neq 0$, and $k_0=+\infty $ if all $a_ l$ vanish. By (1), $h$ is divisible by $w_1^ k$ near $p$ if and only if $k\leq k_0$, so the order of $h$ along $E$ at $p$ is $k_0$. If all $a_ l$ vanish, then $h=0$ on $\Delta $; conversely, if $h$ vanishes near $p$, then all $a_ l$ vanish near the center of $\Delta '$, hence on $\Delta '$. The case $k_0=0$ gives the assertion about $h(0,\cdot )=a_0$, because $a_0$ vanishes identically on $\Delta '$ if and only if it vanishes near the center of $\Delta '$. Finally, let $q=(0,w'_ q)\in E\cap \Delta $. The coordinates $w$ are also coordinates at $q$ with $E=\{ w_1=0\} $ near $q$, and by their formula the coefficients of the expansion of $h$ on a small polydisc centered at $q$ are the restrictions of the same functions $a_ l$. Since $\Delta '$ is connected, $a_ l$ vanishes near $w'_ q$ if and only if it vanishes on $\Delta '$, by the identity theorem. Hence the orders of $h$ along $E$ at $q$ and at $p$ are equal.

(3) By the last assertion of (2), the order of $h$ along $E$ is a locally constant function on $E_{\mathrm{reg}}$, which is connected (corollary 9.3.16). Hence it is constant.

(4) Shrinking $\Delta $, we may assume that $g$ is defined on $\Delta $; let $b_ j$ and $e_ m$ be the coefficients of the expansions of $g$ and $hg$ on $\Delta $. By the Leibniz rule, $e_ m=\sum _{i+j=m}a_ ib_ j$. If $h$ or $g$ has infinite order, then $hg$ vanishes near $p$ and has infinite order by (2). Otherwise let $k$ and $l$ be the orders of $h$ and $g$. Then $e_ m=0$ for $m<k+l$, because each term $a_ ib_ j$ with $i+j<k+l$ has $i<k$ or $j<l$, and $e_{k+l}=a_ kb_ l$. Since $a_ k$ and $b_ l$ are not identically zero and the ring of holomorphic functions on the connected set $\Delta '$ is an integral domain, by the identity theorem, $e_{k+l}\neq 0$. By (2), the order of $hg$ is $k+l$. $\square$