Complex analytic functions
For one variable, the local form of an analytic function ''f''(''z'') near 0 is ''z''''k''''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''2, ..., ''zn''). A Weierstrass polynomial ''W''(''z'') is :''zk'' + ''g''''k''−1''z''''k''−1 + ... + ''g''0 where ''g''''i''(''z''2, ..., ''zn'') is analytic and ''g''''i''(0, ..., 0) = 0. Then the theorem states that for analytic functions ''f'', if :''f''(0, ...,0) = 0, and :''f''(''z'', ''z''2, ..., ''zn'') as a power series has some term only involving ''z'', we can write (locally near (0, ..., 0)) :''f''(''z'', ''z''2, ..., ''zn'') = ''W''(''z'')''h''(''z'', ''z''2, ..., ''zn'') 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''2, ..., ''zn'' 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.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''.Smooth functions
There is a deeper preparation theorem for smooth functions, due to Bernard Malgrange, called the Malgrange preparation theorem. It also has an associated division theorem, named after 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 rings ''A'': for any power series such that not all are in the maximal ideal of ''A'', there is a unique unit ''u'' in and a polynomial ''F'' of the form with (a so-called distinguished polynomial) such that : Since 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''n''·''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 fixedTate algebras
There is also a Weiertrass preparation theorem for Tate algebras : over a complete non-archimedean field ''k''. 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 are Noetherian.See also
* Oka coherence theoremReferences
* *, reprinted in * * * 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 Several complex variables Commutative algebra Theorems in complex analysis