Weierstrass preparation theorem

Remarks

The proof goes through the Weierstrass division theorem. Under the same hypothesis, every $h \in A[\![X]\!]$ can be written uniquely as $h = q f + r$ with $q \in A[\![X]\!]$ and $r$ a polynomial of degree less than $d$. Dividing $X^d$ by $f$ gives $X^d = q f + r$, and one checks that $q$ is a unit and $g = X^d - r$ has its lower coefficients in $\mathfrak m$. The division itself is built by successive approximation modulo powers of $\mathfrak m$, which converges because $A$ is complete. The hypothesis on the coefficients is needed. For $A = k[\![Y]\!]$ and $f = Y$, every coefficient lies in $\mathfrak m = (Y)$, and $f$ is not of the form $u g$. Indeed, modulo $\mathfrak m$ the product $u g$ becomes $\bar u X^d \neq 0$, while $f$ becomes $0$. Applied to $A = k[\![X_1, \dots, X_{n-1}]\!]$, the theorem says that a power series in $n$ variables which is nonzero modulo $(X_1, \dots, X_{n-1})$ is, up to a unit, a polynomial in $X_n$. The original form of the theorem is analytic. Let $f$ be a holomorphic function near $0 \in \C^n$ such that $z_n \mapsto f(0, \dots, 0, z_n)$ has a zero of finite order $d$ at $0$. Then the same factorization holds with convergent series. Here $g$ is a monic polynomial in $z_n$ whose lower coefficients are holomorphic in $z_1, \dots, z_{n-1}$ and vanish at $0$.

Going deeper

For a student

The theorem is the main tool to prove, by induction on $n$, that $k[\![X_1, \dots, X_n]\!]$ over a field $k$ is a Noetherian unique factorization domain. The same holds for the ring of convergent power series in $n$ complex variables. In number theory, for $A = \Z_p$, it shows that a power series in $\Z_p[\![X]\!]$ not divisible by $p$ is a unit times a polynomial, which is used in Iwasawa theory.