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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85e21a26e3b4a4db8088ac6d3c5c9fc4afbf947a)
![{\ displaystyle f \ in {} _ {n-1} {\ mathcal {O}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d27f6660c44981bd610cda118031b77b89e67d1)
![{\ displaystyle f (z_ {1}, \ ldots, z_ {n}): = f (z_ {1}, \ ldots, z_ {n-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5154c3ac323bf92dfefc1b3fc4d4f64deaeff7ce)
![{\ displaystyle {} _ {n-1} {\ mathcal {O}} [z_ {n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4158d16535f3fdeae6c300dba6f85300aad1a427)
![{\ displaystyle {} _ {n} {\ mathcal {O}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85e21a26e3b4a4db8088ac6d3c5c9fc4afbf947a)
![{\ 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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/024eaa65cdbc5776e081def55e060427557bb0f3)
with convergent power series that vanish in 0.
![{\ displaystyle a_ {0}, a_ {1}, \ ldots, a_ {m-1}, a_ {m} \ in {} _ {n-1} {\ mathcal {O}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35c23b53f9a4e0612d7bd10334cdcb7eccfd5fce)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab88b2d6ec02b4d20cf962141c406ddbef2792)
![z_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8228e40cfaa6cb1f163c066ae7054faeff8c7a)
![{\ displaystyle z \ mapsto f (0, \ ldots, 0, z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c11ba838a81260dad0a6ab32f32dc5d85b0dfe)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab88b2d6ec02b4d20cf962141c406ddbef2792)
![z_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8228e40cfaa6cb1f163c066ae7054faeff8c7a)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![{\ displaystyle h \ in {} _ {n-1} {\ mathcal {O}} [z_ {n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4512d93e095ea35be949b6306f709003f6f3181f)
![{\ displaystyle u \ in {} _ {n} {\ mathcal {O}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/282d00648848a00fdc2547eb7e9cbaae7b53a884)
![{\ displaystyle f = u \ cdot h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0200cc4cad9fa74039f96157032763f3111e6567)
Proof idea
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7f66d088f47342474ef5f14a7bb8826280bd42)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![z_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8228e40cfaa6cb1f163c066ae7054faeff8c7a)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![{\ displaystyle 0 <\ delta _ {j} <r_ {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a4f158879a13105bf8e2f3be1354b783ab73e1)
![{\ displaystyle z_ {n} \ mapsto f (z_ {1}, \ ldots, z_ {n-1}, z_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2bcdb2d73d1879c250acd5dc425381d83b4703b)
![{\ displaystyle (z_ {1}, \ ldots, z_ {n-1}) \ in \ Delta (0, \ delta _ {1}, \ ldots, \ delta _ {n-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66bf2844b439906fcd4354437bfa8c348be793f3)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![{\ displaystyle \ Delta (0; \ delta _ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce66e0104ff72b4cdddf6b3f2bcd30d8834654e1)
![{\ displaystyle \ varphi _ {1} (z_ {1}, \ ldots, z_ {n-1}), \ ldots, \ varphi _ {m} (z_ {1}, \ ldots, z_ {n-1} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d044a52fc1b46e181cd358e9a17b5688fe91998c)
![{\ displaystyle h (z_ {1}, \ ldots, z_ {n}): = \ prod _ {k = 1} ^ {m} (z_ {n} - \ varphi _ {k} (z_ {1}, \ ldots, z_ {n-1}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ef30f93c25e4fa5138825b060f7f53742e36b3)
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.
![u](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
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 z_ {1} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84eb9ddefc80e2c5a81a8ef85218afbd6112a46f)
![{\ displaystyle f (z_ {1}) = u (z_ {1}) z_ {1} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82383e9f6080ee2e441248f703e8c3ae03720087)
![u](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![{\ displaystyle u (z_ {1}) z_ {1} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c999e92e25729278d46e9ac4c563ccc643369faa)
![u](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ab88b2d6ec02b4d20cf962141c406ddbef2792)
![z_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8228e40cfaa6cb1f163c066ae7054faeff8c7a)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![{\ displaystyle f (z_ {1}, \ ldots, z_ {n}) = u (z_ {1}, \ ldots, z_ {n}) \ cdot (z_ {n} -a (z_ {1}, \ ldots, z_ {n-1}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d5a9680439fcc26a0dc0e6abcae11f7ac28b54)
with a holomorphic function . Da , in a neighborhood of 0
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ displaystyle u (0) \ not = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e10b1e2c9b6890553f3004318ecb3da1a92d436a)
-
.
See also
Individual evidence
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^ Wolfgang Ebeling: Theory of functions, differential topology and singularities , Vieweg-Verlag (2001), ISBN 978-3-528-03174-9 , Theorem 2.1
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^ Gunning - Rossi : Analytic functions of several complex variables . Prentice-Hall 1965, chap. II.B, Theorem 2 (Weierstrass Preparation Theorem)
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↑ Wolfgang Ebeling: Theory of functions, differential topology and singularities , Vieweg-Verlag (2001), ISBN 978-3-528-03174-9 , comment 2.3
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^ Gunning-Rossi: Analytic functions of several complex variables . Prentice-Hall 1965, chap. II.B, page 70