# Weierstrass preparation theorem

The following theorem is known as the Weierstrass preparation theorem, though sometimes that name is reserved for the corollary and this theorem is then known as the Weierstrass division theorem.

In the following we use the standard notation for coordinates  in ${\mathbb{C}}^{n}$ that $z=(z_{1},\ldots,z_{n})=(z^{\prime},z_{n})$. That is $z^{\prime}$ is the first $n-1$ coordinates.

###### Theorem.

Let $f\colon{\mathbb{C}}^{n}\to{\mathbb{C}}$ be a function analytic  in a neighbourhood $U$ of the origin such that $\frac{f(0,z_{n})}{z_{n}^{m}}$ extends to be analytic at the origin and is not zero at the origin for some positive integer $m$ (in other words, as a function of $z_{n}$, the function has a zero of order $m$ at the origin). Then there exists a polydisc $D\subset U$ such that every function $g$ holomorphic and bounded  in $D$ can be written as

 $g=qf+r,$

where $q$ is an analytic function and $r$ is a polynomial in the $z_{n}$ variable of degree less then $m$ with the coefficients being holomorphic functions in $z^{\prime}$. Further there exists a constant $C$ independent of $g$ such that

 $\sup_{z\in D}\lvert q(z)\rvert\leq C\sup_{z\in D}\lvert g(z)\rvert.$

The representation $g=qf+r$ is unique. Finally the coefficients of the power series  expansions of $q$ and $r$ are finite linear combinations  of the coefficients of the power series of $g$.

Note that $r$ is not necessarily a Weierstrass polynomial.

###### Corollary.

Let $f$ be as above, then there is a unique representation of $f$ as $f=hW$, where $h$ is analytic in a neighbourhood of the origin and $h(0)\not=0$ and $W$ being a Weierstrass polynomial.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Weierstrass preparation theorem WeierstrassPreparationTheorem 2013-03-22 15:04:28 2013-03-22 15:04:28 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 32B05 Weierstrass division theorem WeierstrassPolynomial