Weierstrass preparation theorem

In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of 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, and whose coefficients 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 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 z^kg(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, ..., z_n). A Weierstrass polynomial W(z) is

z^k + g_k-1z^k-1 + ... + g₀

where g_i(z₂, ..., z_n) is analytic and g_i(0, ..., 0) = 0.

Then the theorem states that for analytic functions f, if f(0, ...,0) = 0, but f(z, z₂, ..., z_n) as a power series has some term not involving z, we can write (locally near (0, ..., 0))

f(z, (0, ..., 0)) (0, ..., 0)) ) = W(z)h(z, z₂, ..., z_n)

with h analytic and h(0, ..., 0) = 0, and W a Weierstrass polynomial.

This has the immediate consequence that the set of zeroes 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, ..., z_n 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.

There is a deeper preparation theorem for smooth functions, due to Malgrange.