# The Weierstrass Preparation and Division Theorems.

Speaker : Samrat Sen.

Date : 10 February, 2015 (Tuesday)

Time : 09:15 pm – 10:15 pm

Venue : Lecture Hall I, Department of Mathematics

Abstract: The Weierstrass Preparation and Division Theorems are the main tools to understand the structure of the local ring consisting of the germs of holomorphic functions in several variables, at a given point $P$. According to the Weierstrass Preparation Theorem, any non-zero germ is, up to a non-singular linear change of coordinates and multiplication by a germ of a function not zero at $P$, a Weierstrass Polynomial in one variable of certain degree. On the other hand, the Weierstrass Division theorem says that, given any germ $f$ and a suitable one-variable Weierstrass Polynomial $g$ of degree $k>0$, there exists a unique pair of germs $h$ and $j$ such that $f=g \cdot h+j$, where $j$ is a Weierstrass Polynomial of degree less than $k$. We’ll prove both of the theorems using elementary several complex variables techniques in the seminar and lastly we’ll see that these are the extended forms of the Implicit Function Theorem and the Division Algorithm on polynomial rings respectively.

Area: Complex Analysis, Several Complex Variables