Weierstrasse preparatory sentence

The Weierstrass preparation set is a mathematical theorem from the function theory of several variables. It establishes a connection between zeros of power series and Weierstraß polynomials .

Introduction and formulation of the sentence

Let it denote the ring of convergent power series around 0. Each can be regarded as an element of by means of the definition . In particular, that is polynomial in included. Therefore one can speak of a polynomial degree . This is especially true for Weierstrass polynomials, that is, polynomials of the form ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle f\in {}_{n-1}{\mathcal {O}}}$ ${\displaystyle f(z_{1},\ldots ,z_{n}):=f(z_{1},\ldots ,z_{n-1})}$ ${\displaystyle {}_{n}{\mathcal {O}}}$ ${\displaystyle {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle {}_{n}{\mathcal {O}}}$

${\displaystyle z_{n}^{m}+a_{m-1}(z_{1},\ldots ,z_{n-1})\cdot z_{n}^{m-1}+\ldots +a_{1}(z_{1},\ldots ,z_{n-1})\cdot z_{n}+a_{0}(z_{1},\ldots ,z_{n-1})}$

with convergent power series that vanish in 0. ${\displaystyle a_{0},a_{1},\ldots ,a_{m-1},a_{m}\in {}_{n-1}{\mathcal {O}}}$

A power series is called regular of order if the holomorphic function in 0 has a zero of order . ${\displaystyle f\in {}_{n}{\mathcal {O}}}$ ${\displaystyle z_{n}}$ ${\displaystyle m}$ ${\displaystyle z\mapsto f(0,\ldots ,0,z)}$ ${\displaystyle m}$

With these conceptualizations, the following Weierstrasse preparatory sentence applies.

• Let be a convergent power series that is regular in order . Then there is a uniquely determined Weierstraß polynomial of degree and a uniquely determined unit with . ${\displaystyle f\in {}_{n}{\mathcal {O}}}$ ${\displaystyle z_{n}}$ ${\displaystyle m}$ ${\displaystyle h\in {}_{n-1}{\mathcal {O}}[z_{n}]}$ ${\displaystyle m}$ ${\displaystyle u\in {}_{n}{\mathcal {O}}}$ ${\displaystyle f=u\cdot h}$

Proof idea

${\displaystyle f}$converges on a suitable polycircle . Since in is of regular order , one finds such that the function has exactly zeros in the circle for every fixed one . Let these be denoted by, with repetitions occurring for multiple zeros. One multiplies ${\displaystyle \Delta (0;r_{1},\ldots ,r_{n})}$ ${\displaystyle f}$ ${\displaystyle z_{n}}$ ${\displaystyle m}$ ${\displaystyle 0<\delta _{j}<r_{j}}$ ${\displaystyle z_{n}\mapsto f(z_{1},\ldots ,z_{n-1},z_{n})}$ ${\displaystyle (z_{1},\ldots ,z_{n-1})\in \Delta (0,\delta _{1},\ldots ,\delta _{n-1})}$ ${\displaystyle m}$ ${\displaystyle \Delta (0;\delta _{n})}$ ${\displaystyle \varphi _{1}(z_{1},\ldots ,z_{n-1}),\ldots ,\varphi _{m}(z_{1},\ldots ,z_{n-1})}$

${\displaystyle h(z_{1},\ldots ,z_{n}):=\prod _{k=1}^{m}(z_{n}-\varphi _{k}(z_{1},\ldots ,z_{n-1}))}$

off, a Weierstrass polynomial is obtained that does what is required.

comment

The name preparation sentence comes from the fact that the power series is prepared for the investigation of its roots . Since the factor does not vanish as a unit in a neighborhood of 0, the zeros in such a neighborhood are the same as those of the Weierstrass polynomial. ${\displaystyle u}$

For , that is, for holomorphic functions of a variable, the Weierstraß polynomial must be the normalized monomial . It is then with a holomorphic function that does not vanish in 0. The preparatory theorem therefore generalizes the fact that a holomorphic function of a variable with -fold zeros in 0 can be written as with a holomorphic function that does not vanish in 0 , to dimensions. ${\displaystyle n=1}$ ${\displaystyle z_{1}^{m}}$ ${\displaystyle f(z_{1})=u(z_{1})z_{1}^{m}}$ ${\displaystyle u}$ ${\displaystyle m}$ ${\displaystyle u(z_{1})z_{1}^{m}}$ ${\displaystyle u}$ ${\displaystyle n}$

In order to classify the proposition, it should be mentioned that a proposition about implicit functions can easily be derived from it . If in regular is of the first order, then according to the preparatory sentence has the form ${\displaystyle f\in {}_{n}{\mathcal {O}}}$ ${\displaystyle z_{n}}$ ${\displaystyle f}$

${\displaystyle f(z_{1},\ldots ,z_{n})=u(z_{1},\ldots ,z_{n})\cdot (z_{n}-a(z_{1},\ldots ,z_{n-1}))}$

with a holomorphic function . Da , in a neighborhood of 0 ${\displaystyle a}$ ${\displaystyle u(0)\not =0}$

${\displaystyle f(z_{1},\ldots ,z_{n})=0\Longleftrightarrow z_{n}=a(z_{1},\ldots ,z_{n-1})}$.

See also

• Weierstrass division set

Individual evidence

1. ^ Wolfgang Ebeling: Theory of functions, differential topology and singularities , Vieweg-Verlag (2001), ISBN 978-3-528-03174-9 , Theorem 2.1
2. ^ Gunning - Rossi : Analytic functions of several complex variables . Prentice-Hall 1965, chap. II.B, Theorem 2 (Weierstrass Preparation Theorem)
3. ↑ Wolfgang Ebeling: Theory of functions, differential topology and singularities , Vieweg-Verlag (2001), ISBN 978-3-528-03174-9 , comment 2.3
4. ^ Gunning-Rossi: Analytic functions of several complex variables . Prentice-Hall 1965, chap. II.B, page 70