- Groebner bases

###### Options

# Groebner bases

Author

Koh, Diane Jael Mei Lin

Supervisor

Teo, Kok Ming

Abstract

Consider a polynomial ring k[χ] in one indeterminate over a field k. Given a polynomial in k[χ], one is able to determine whether that polynomial lies in a given ideal I C k[χ] by applying the Division Algorithm in one variable. The condition r = 0 is necessary and sufficient for membership to the ideal. However, this is not the case for the polynomial ring k[χl...,χn] because remainders generated by the Division Algorithm in n variables are not unique.

By the Hilbert Basis Theorem, any given ideal J C k[χl...,χn] has a finite generating set, that is, J = 〈gl,...,gt〉. Then gl,...,gt are a basis of J. They are also known as generators of J. A Groebner basis of J is a special basis whereby the remainder on division by the generators is unique with respect to some fixed ordering. We can then determine ideal membership by checking the remainder.

Besides studying the basic theory of Groebner bases, we will look at how we can construct them using the Buchberger's Algorithm and S-polynomials.. Groebner bases also have several applications. The two applications that we are going to study are ideal membership and solving systems of polynomial equations.

By the Hilbert Basis Theorem, any given ideal J C k[χl...,χn] has a finite generating set, that is, J = 〈gl,...,gt〉. Then gl,...,gt are a basis of J. They are also known as generators of J. A Groebner basis of J is a special basis whereby the remainder on division by the generators is unique with respect to some fixed ordering. We can then determine ideal membership by checking the remainder.

Besides studying the basic theory of Groebner bases, we will look at how we can construct them using the Buchberger's Algorithm and S-polynomials.. Groebner bases also have several applications. The two applications that we are going to study are ideal membership and solving systems of polynomial equations.

Date Issued

2003

Call Number

QA251.3 Koh

Date Submitted

2003