Gröbner Bases, Elimination, and Generic Initial Ideals

Gröbner bases are a tool for doing explicit calculations in a polynomial ring over a field. Gröbner bases can be used to calculate two specific computational problems:
1. (Membership Problem) Given an element f in a polynomial ring, do we know whether f is in a particular ideal I of the polynomial ring?
2. (Elimination) Given a finite set of generators for an ideal I⊂k[x_1,… ,x_n], can we find a finite set of generators for I∩ k[x_(r + 1),… ,x_n], 1≤ r≤ n-1?

Even though computational problems are an important application of Gröbner bases, they can also be used to simplify some very well-known theorems such as Hilbert's Basis Theorem and the natural isomorphism between the ring of elementary symmetric functions over a field and the symmetric polynomials. Gröbner bases are also used for theoretical problems, although this thesis will focus on the computational problems.

In this thesis we will define a Gröbner basis and prove several theorems related to the topic such as Dickson's Lemma, the Division Algorithm, Buchberger's Criterion, the Elimination Theorem, and the Existence of Generic Initial Ideals. Examples are also provided throughout the thesis for better understanding of the topic.