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 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 R as u·w, where u is a 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. For one variable, the local form of an analytic function f(z) near 0 is zkh(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, z2, ..., zn). A Weierstrass polynomial W(z) is zk + gk−1zk−1 + ... + g0 where gi(z2, ..., zn) is analytic and gi(0, ..., 0) = 0. Then the theorem states that for analytic functions f, if f(0, ...,0) = 0, and f(z, z2, ..., zn) as a power series has some term only involving z, we can write (locally near (0, ..., 0)) f(z, z2, ..., zn) = W(z)h(z, z2, ..., 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 z2, ..., 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. 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.