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

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, ..., 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.

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 = z^N for the least N that gives an h not zero at the origin; the desired Weierstrass polynomial is then z^N + 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 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.

p-adic Analogue

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 πⁿ·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 π is a fixed uniformizer.

References

• Siegel, C. L. (1969), "Zu den Beweisen des Vorbereitungssatzes von Weierstrass", Number Theory and Analysis (Papers in Honor of Edmund Landau), New York: Plenum, pp. 297–306, MR0268402, reprinted in Siegel, Carl Ludwig (1979), Chandrasekharan, K.; Maass., H. (eds.), Gesammelte Abhandlungen. Band IV, Berlin-New York: Springer-Verlag, pp. 1–8, ISBN 0-387-09374-5, MR0543842
• Solomentsev, E.D. (2001) [1994], "Weierstrass theorem", Encyclopedia of Mathematics, EMS Press
• Stickelberger, L. (1887), "Ueber einen Satz des Herrn Noether", Mathematische Annalen, 30 (3): 401–409, doi:10.1007/BF01443952
• Weierstrass, K. (1895), Mathematische Werke. II. Abhandlungen 2, Berlin: Mayer & Müller, pp. 135–142 reprinted by Johnson, New York, 1967.