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

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s