Weierstrass division theorem

Remarks

The same statement holds for germs of holomorphic functions at $0$, since these form a ring isomorphic to $\mathcal{O}_n$. One proof first treats a divisor $f = P$ that is a Weierstrass polynomial of degree $d$. On a small polydisc $\abs{z'} < \varepsilon$, $\abs{z_n} < \delta$, the polynomial $P(z', \cdot)$ has its $d$ roots inside $\abs{z_n} < \delta$ for each $z'$. The Cauchy integral $$q(z', z_n) = \frac{1}{2\pi i} \oint_{\abs{w} = \delta} \frac{g(z', w)}{P(z', w)\,(w - z_n)}\, dw$$ defines a holomorphic $q$, and the Cauchy formula for $g$ gives $$r = g - q P = \frac{1}{2\pi i} \oint_{\abs{w} = \delta} \frac{g(z', w)}{P(z', w)} \cdot \frac{P(z', w) - P(z', z_n)}{w - z_n}\, dw.$$ The last quotient is a polynomial in $z_n$ of degree $d - 1$, so $r$ is a polynomial in $z_n$ of degree less than $d$. A general $f$ regular of order $d$ is a unit times a Weierstrass polynomial. This can be shown directly through the symmetric functions of the roots of $f(z', \cdot)$, and it reduces the general case to the first one. Example: for $n = 1$ and $f = z^d$, the theorem says $g = q z^d + r$, where $r$ is the Taylor polynomial of $g$ of degree $d - 1$. The regularity hypothesis is needed: for $f = z_1$ in $\mathcal{O}_2$ and any $d$, the series $g = z_2^d$ is not of the form $q z_1 + r$ with $\deg_{z_2} r < d$, since setting $z_1 = 0$ would give $z_2^d = r(0, z_2)$. Conversely, applying the theorem to $g = z_n^d$ gives the Weierstrass preparation theorem: $f$ is a unit times the Weierstrass polynomial $z_n^d - r$. A linear change of coordinates makes any nonzero $f \in \mathcal{O}_n$ regular in $z_n$ of some order, so the hypothesis is mild.

Going deeper

For a student

Division is the engine behind the algebra of the ring $\mathcal{O}_n$. It implies the preparation theorem, and with it one proves by induction on $n$ that $\mathcal{O}_n$ is Noetherian and factorial. It also shows that $\mathcal{O}_n/(f)$ is a free module of rank $d$ over $\mathcal{O}_{n-1}$ when $f$ is regular of order $d$. This is the algebraic form of the local parametrization of hypersurfaces.