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Weierstrass preparation theorem

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Overview

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

In one variable, a holomorphic function ff, locally holomorphic around z 0z_0, can be represented as f(z)=(zz 0) nu(z)f(z)=(z-z_0)^n u(z) where u(z 0)0u(z_0)\neq 0, uu is holomorphic and nn 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 n2n\geq 2; then we separate the first (n1)(n-1) complex coordinates z=(z 1,,z n1)z = (z_1,\ldots,z_{n-1}) and the nn-th coordinate which will be denoted by ww. We consider an analytic function f=f(z 1,,z n1,w)f = f(z_1,\ldots, z_{n-1},w) vanishing at origin f(0,,0)=0f(0,\ldots, 0)=0, and such that it is not identically zero on the ww-axis.

Weierstrass polynomial

The Weierstrass polynomial of ww is a polynomial of the form

w d+a 1(z)w d1++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 dd is called the degree of the Weierstrass polynomial.

Weierstrass preparation theorem in n\mathbb{C}^n

Let ff be a function which is holomorphic in some neighborhood of origin 0 n0\in\mathbb{C}^n and not identically equal to zero on the ww-axis. Then there is a neighborhood of origin such that ff is uniquely representable in the form

f=Ph f = P\cdot h

where PP is a Weierstrass polynomial of degree dd of ww and h(0)0h(0) \neq 0.

Weierstrass division theorem

Let 𝒪 n,a\mathcal{O}_{n,a} be the local ring of germs of holomorphic functions at a na\in\mathbb{C}^n and 𝒪 n:=𝒪 n,0\mathcal{O}_n:=\mathcal{O}_{n,0}. Let g=g(z,w)𝒪 n1[w]g=g(z,w)\in\mathcal{O}_{n-1}[w] be a Weierstrass polynomial of degree kk of ww. Then every holomorphic function f𝒪 nf\in\mathcal{O}_n can be represented as

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

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

As a corollary, if another function hh vanishes on the zero set of ff, then ff divides hh in 𝒪 n\mathcal{O}_n.

Ingredients of proofs

Weierstrass’s original result considered the ring of holomorphic functions, and therefore used analytic methods such as the residue theorem? and Cauchy integral formula are used. Analogous results are now known in the ring of formal power series, as well as for power series over non-Archimedean fields; in these settings, the proof can be made fully algebraic.

References

Named after Karl Weierstraß.

category: analysis, geometry

Last revised on September 2, 2020 at 08:22:36. See the history of this page for a list of all contributions to it.