Gröbner Basis with applications

A basis for an ideal is such that every element in the ideal can be expressed as a linear combination of the basis. With a Gröbner Basis, every polynomial can be expressed as a linear combination of the basis with a unique remainder. In recent years, there has been a growing study in such classical...

Full description

Saved in:
Bibliographic Details
Main Author: Zhang, Eric Boyuan
Other Authors: Wu Guohua
Format: Final Year Project
Language:English
Published: Nanyang Technological University 2020
Subjects:
Online Access:https://hdl.handle.net/10356/139095
Tags: Add Tag
No Tags, Be the first to tag this record!
Institution: Nanyang Technological University
Language: English
Description
Summary:A basis for an ideal is such that every element in the ideal can be expressed as a linear combination of the basis. With a Gröbner Basis, every polynomial can be expressed as a linear combination of the basis with a unique remainder. In recent years, there has been a growing study in such classical cases with its applications to areas outside of mathematics. We study the concept of a Gröbner Basis and analyse fundamental theorems such as Dickson’s lemma and Hilbert Basis Theorem that are necessary for the construction of the Gröbner bases and improved algorithms to produce such a basis. We provide an alternative perspective for some fundamental theorems as well as the F4 algorithm which reduces the computational complexity of Buchberger’s algorithm. Further, we explore applications of Gröbner bases to Algebraic Geometry and Commutative Algebra.