Editing Weierstrass preparation theorem (section)

==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>