Weierstrass preparation theorem
Encyclopedia
In mathematics
, the Weierstrass preparation theorem is a tool for dealing with analytic function
s 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 coefficient
s 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 zkh(z) where h(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
where
is analytic and
Then the theorem states that for analytic functions f, if
but
as a power series has some term only 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 a deeper preparation theorem for smooth function
s, due to Bernard Malgrange
, called the Malgrange preparation theorem
. It also has an associated division theorem, named after John Mather.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Weierstrass preparation theorem is a tool for dealing with analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
s of several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
, at a given point P. It states that such a function is, up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
multiplication by a function not zero at P, a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
in one fixed variable z, which is monic, and whose coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s 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
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R as u·w, where u is a unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
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 zkh(z) where h(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
- 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,
but
- 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 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
Polynomial remainder theorem
In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division. It states that the remainder of a polynomial f\, divided by a linear divisor x-a\, is equal to f \,.- Example :...
on the resulting Weierstrass polynomial.
There is a deeper preparation theorem for smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s, due to Bernard Malgrange
Bernard Malgrange
Bernard Malgrange is a French mathematician who works on differential equations and singularity theory.He proved the Ehrenpreis–Malgrange theorem and the Malgrange preparation theorem....
, called the Malgrange preparation theorem
Malgrange preparation theorem
In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by .-Statement of Malgrange preparation theorem:...
. It also has an associated division theorem, named after John Mather.