There are also a number of variants of the theorem, that extend the idea of factorization in some ring R as u.w, where u is a unit and w is some sort of distinguished Weierstrass polynomial. C.L. 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.
For one variable, the local form of an analytic function f(z) near 0 is zkg(z) where g(0) is not 0, and k is the order of zero of f at 0. This is the result the preparation theorem generalises. We pick out one variable z, which we may assume is first, and write our complex variables as (z, z2, ..., zn). A Weierstrass polynomial W(z) is
is analytic and
Then the theorem states that for analytic functions f, if
as a power series has some term not involving z, we can write (locally near (0, ..., 0))
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 value of z and then solving W(z). The corresponding values of z2, ..., zn 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.
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. This is equivalent to the preparation theorem, since the Weierstrass factorization of f may be obtained by applying the division theorem for g = zN for the least N that gives an h not zero at the origin; the desired Weierstrass polynomial is then zN + j/h. For the other direction, we use the preparation theorem on g, and the normal polynomial remainder theorem on the resulting Weierstrass polynomial.
There is an analogous result, also referred to as the Weierstrass preparation theorem, for power series rings over the ring of integers in a p-adic field; namely, a power series f(z) can always be uniquely factored as (pi)^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 term each in the maximal ideal), and pi is a fixed uniformizer.