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Weierstrass preparation theorem

Overview

We are interested in the local structure of zeros of analytic functions in ${ℂ}^n$ as well as in analogues, e.g. in rigid analytic geometry.

In one variable, a holomorphic function $f$, locally holomorphic around $z_0$, can be represented as $f(z)=(z-z_0{)}^{n}u(z)$ where $u(z_0)\ne 0$, $u$ is holomorphic and $n$ is a nonnegative integer; therefore the solution set is discrete. In many variables, these zero sets are more complicated but far from arbitrary; in fact the analytic sets are often pretty close to algebraic varieties: for example, analytic subsets of the projective space are algebraic.

The Weierstrass preparation theorem and related facts (Weierstrass division theorem and Weierstrass formula) provide the most basic relations between polynomials and holomorphic functions.

Let $n\ge 2$; then we separate the first $(n-1)$ complex coordinates $z=(z_1,\dots ,z_{n-1})$ and the $n$-th coordinate which will be denoted by $w$. We consider an analytic function $f=f(z_1,\dots ,z_{n-1},w)$ vanishing at origin $f(0,\dots ,0)=0$, and such that it is not identically zero on the $w$-axis. Let ${𝒪}_{n,a}$ be the local ring of germs of holomorphic functions at $a\in {ℂ}^n$ and ${𝒪}_n:={𝒪}_{n,0}$.

Weierstrass polynomial

The Weierstrass polynomial of $w$ is a polynomial of the form

$$w^d+a_1(z)w^{d-1}+\dots +a_d(z),a_i(0)=0.$$w^d + a_1(z) w^{d-1}+\ldots+a_d(z),\,\,\,\,\,a_i(0)=0.

The integer $d$ is called the degree of the Weierstrass polynomial.

Weierstrass preparation theorem in ${ℂ}^n$

Let $f$ be a function which is holomorphic in some neighborhood of origin $0\in {ℂ}^n$ and not identically equal to zero on the $w$-axis. Then there is a neighborhood of origin such that $f$ is uniquely representable in the form

$$f=P\cdot h$$f = P\cdot h

where $P$ is a Weierstrass polynomial of degree $d$ of $w$ and $h(0)\ne 0$.

Weierstrass division theorem

Let $g=g(z,w)\in {𝒪}_{n-1}[w]$ be a Weierstrass polynomial of degree $k$ of $w$. Then every holomorphic function $f\in {𝒪}_n$ can be represented as

$$f=g\cdot h+r$$f = g\cdot h+r

where $r=r(z,w)$ is a polynomial of degree $<k$.

As a corollary, if another function $h$ vanishes on the zero set of $f$, then $f$ divides $h$ in ${𝒪}_n$.

Ingredients of proofs

Weierstrass used analytic methods to prove the theorem; in fact the residue theorem? and Cauchy integral formula? are used. However, much later a fully algebraic proof has been found, and it allows generalizations to much wider setups, not only over complex numbers.

category: analysis, geometry