Editing Weierstrass preparation theorem

{{Use American English|date = March 2019}}
{{Short description|Local theory of several complex variables}}
In [[mathematics]], the '''Weierstrass preparation theorem''' is a tool for dealing with [[analytic function]]s of [[Function of several complex variables|several complex variables]], at a given point ''P''. It states that such a function is, [[up to]] multiplication by a function not zero at ''P'', a [[polynomial]] in one fixed variable ''z'', which is [[monic polynomial|monic]], and whose [[coefficient]]s of lower degree terms are analytic functions in the remaining variables and zero at ''P''.

There are also a number of variants of the theorem, that extend the idea of factorization in some [[ring (mathematics)|ring]] ''R'' as ''u''·''w'', where ''u'' is a [[Unit (ring theory)|unit]] and ''w'' is some sort of distinguished '''Weierstrass polynomial'''. [[Carl Siegel]] has disputed the attribution of the theorem to [[Weierstrass]], saying that it occurred under the current name in some of late nineteenth century ''Traités d'analyse'' without justification.

==Complex analytic functions==
For one variable, the local form of an analytic function ''f''(''z'') near 0 is ''z''<sup>''k''</sup>''h''(''z'') where ''h''(0) is not 0, and ''k'' is the order of the zero of ''f'' at 0. This is the result that the preparation theorem generalises. 
We pick out one variable ''z'', which we may assume is first, and write our complex variables as (''z'', ''z''<sub>2</sub>, ..., ''z<sub>n</sub>''). A Weierstrass polynomial ''W''(''z'') is

:''z<sup>k</sup>'' + ''g''<sub>''k''&minus;1</sub>''z''<sup>''k''&minus;1</sup> + ... + ''g''<sub>0</sub>

where ''g''<sub>''i''</sub>(''z''<sub>2</sub>, ..., ''z<sub>n</sub>'') is analytic and ''g''<sub>''i''</sub>(0, ..., 0) = 0.

Then the theorem states that for analytic functions ''f'', if

:''f''(0, ...,0) = 0,

and

:''f''(''z'', ''z''<sub>2</sub>, ..., ''z<sub>n</sub>'')

as a [[power series]] has some term only involving ''z'', we can write (locally near (0, ..., 0))

:''f''(''z'', ''z''<sub>2</sub>, ..., ''z<sub>n</sub>'') = ''W''(''z'')''h''(''z'', ''z''<sub>2</sub>, ..., ''z<sub>n</sub>'')

with ''h'' analytic and ''h''(0, ..., 0) not 0, and ''W ''a Weierstrass polynomial.

This has the immediate consequence that the set of zeros of ''f'', near (0, ..., 0), can be found by fixing any small values of ''z''<sub>2</sub>, ..., ''z<sub>n</sub>'' and then solving the equation ''W(z)=0''. The corresponding values of ''z'' form a number of continuously-varying ''branches'', in number equal to the degree of ''W'' in ''z''. In particular ''f'' cannot have an isolated zero.

===Division theorem===
A related result is the '''Weierstrass division theorem''', which states that if ''f'' and ''g'' are analytic functions, and ''g'' is a Weierstrass polynomial of degree ''N'', then there exists a unique pair ''h'' and ''j'' such that ''f'' = ''gh'' + ''j'', where ''j'' is a polynomial of degree less than ''N''. In fact, many authors prove the Weierstrass preparation as a corollary of the division theorem. It is also possible to prove the division theorem from the preparation theorem so that the two theorems are actually equivalent.<ref>{{citation|title=Analytische Stellenalgebren|author1=Grauert, Hans|author1-link=Hans Grauert|author2=Remmert, Reinhold|author2-link=Reinhold Remmert|publisher=Springer|language=German|page=43|doi=10.1007/978-3-642-65033-8|year=1971|isbn=978-3-642-65034-5}}</ref>

===Applications===
The Weierstrass preparation theorem can be used to show that the ring of germs of analytic functions in ''n'' variables is a Noetherian ring, which is also referred to as the ''Rückert basis theorem''.<ref>{{citation|author=Ebeling, Wolfgang|title=Functions of Several Complex Variables and Their Singularities|publisher=[[American Mathematical Society]]|year=2007|location=Proposition 2.19}}</ref>

==Smooth functions==
There is a deeper preparation theorem for [[smooth function]]s, due to [[Bernard Malgrange]], called the [[Malgrange preparation theorem]]. It also has an associated division theorem, named after [[John Mather (mathematician)|John Mather]].

==Formal power series in complete local rings==

There is an analogous result, also referred to as the Weierstrass preparation theorem, for the ring of [[formal power series]] over [[complete local ring]]s ''A'':<ref>{{citation|author=Nicolas Bourbaki|author-link=Nicolas Bourbaki|title=Commutative algebra|location=chapter VII, §3, no. 9, Proposition 6|year=1972|publisher=Hermann}}</ref> for any power series <math>f = \sum_{n=0}^\infty a_n t^n \in A[[t]]</math> such that not all <math>a_n</math> are in the [[maximal ideal]] <math>\mathfrak m</math> of ''A'', there is a unique [[unit (ring theory)|unit]] ''u'' in <math>A[[t]]</math> and a polynomial ''F'' of the form <math>F=t^s + b_{s-1} t^{s-1} + \dots + b_0</math> with <math>b_i \in \mathfrak m</math> (a so-called distinguished polynomial) such that
:<math>f = uF.</math>
Since <math>A[[t]]</math> is again a complete local ring, the result can be iterated and therefore gives similar factorization results for formal power series in several variables.

For example, this applies to the ring of integers in a [[p-adic]] field. In this case the theorem says that a power series ''f''(''z'') can always be uniquely factored as π<sup>''n''</sup>·''u''(''z'')·''p''(''z''), where ''u''(''z'') is a unit in the ring of power series, ''p''(''z'') is a [[distinguished polynomial]] (monic, with the coefficients of the non-leading terms each in the maximal ideal), and π is a fixed [[uniformizer]].

An application of the Weierstrass preparation and division theorem for the ring <math>\mathbf Z_p[[t]]</math> (also called [[Iwasawa algebra]]) occurs in [[Iwasawa theory]] in the description of finitely generated modules over this ring.<ref>{{citation|author=Lawrence Washington|author-link=Lawrence C. Washington|title=Introduction to cyclotomic fields|publisher=Springer|year=1982|location=Theorem 13.12}}</ref>

There exists a non-commutative version of Weierstrass division and preparation, with ''A'' being a not necessarily commutative ring, and with formal skew power series in place of formal power series.<ref>{{cite journal |author=Otmar Venjakob |title=A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory |journal=J. Reine Angew. Math. |year=2003 |volume=2003 |issue=559 |pages=153–191 |doi=10.1515/crll.2003.047 |s2cid=14265629 |url=https://www.degruyter.com/document/doi/10.1515/crll.2003.047/pdf |accessdate=2022-01-27|arxiv=math/0204358 }} Theorem 3.1, Corollary 3.2</ref>

==Tate algebras==

There is also a Weierstrass preparation theorem for [[Tate algebra]]s
:<math>T_n(k) = \left  \{ \sum_{\nu_1, \dots, \nu_n \ge 0} a_{\nu_1, \dots, \nu_n} X_1^{\nu_1} \cdots X_n^{\nu_n}, |a_{\nu_1, \dots, \nu_n}| \to 0 \text{ for } \nu_1 + \dots +\nu_n \to \infty \right \}</math>
over a complete [[non-archimedean field]] ''k''.<ref>{{citation|author1=Bosch, Siegfried|author-link=Siegfried Bosch|author2=Güntzer, Ulrich|author3=Remmert, Reinhold|author3-link=Reinhold Remmert|title=Non-archimedean analysis|location=Chapters 5.2.1, 5.2.2|publisher=Springer|year=1984}}</ref>  
These algebras are the basic building blocks of [[rigid geometry]]. One application of this form of the Weierstrass preparation theorem is the fact that the rings <math>T_n(k)</math> are [[Noetherian ring|Noetherian]].

== See also ==
* [[Oka coherence theorem]]
==References==
{{Reflist}}
*{{citation|last=Lewis|first=Andrew|title=Notes on Global Analysis| url=http://www.mast.queensu.ca/~andrew/teaching/math942/}}
*{{citation|mr=0268402
|last=Siegel|first= C. L.
|chapter=Zu den Beweisen des Vorbereitungssatzes von Weierstrass |year=1969|title=  Number Theory and Analysis (Papers in Honor of Edmund Landau) |pages= 297–306 |publisher=Plenum|location= New York }}, reprinted in {{citation|mr=0543842|last= Siegel|first= Carl Ludwig |title=Gesammelte Abhandlungen. Band IV|editor-first= K. |editor-last=Chandrasekharan|editor2-first=H.|editor2-last= Maass.|publisher= Springer-Verlag|location=Berlin-New York|year= 1979|pages=1–8| isbn= 0-387-09374-5}}
*{{springer|id=W/w097510|title=Weierstrass  theorem|first=E.D.|last= Solomentsev}}
*{{citation|title= Ueber einen Satz des Herrn Noether
|doi	=10.1007/BF01443952
|journal	=Mathematische Annalen
|volume= 30|issue= 3 |year= 1887|pages=401–409
|first=L.|last= Stickelberger|s2cid	=121360367
|url	=https://zenodo.org/record/1599614
}}
*{{citation|first=K.|last=Weierstrass|  title=Mathematische Werke. II. Abhandlungen 2|pages= 135–142|publisher= Mayer & Müller|location= Berlin|year= 1895}} reprinted by Johnson, New York, 1967.
== External links ==
*{{cite web |last1=Lebl |first1=Jiří |title=Weierstrass Preparation and Division Theorems. (2021, September 5). |url=https://math.libretexts.org/@go/page/74245 |website=LibreTexts|date=6 July 2021 }}
[[Category:Several complex variables]]
[[Category:Commutative algebra]]
[[Category:Theorems in complex analysis]]