Abstract

An ideal I in a polynomial ring k[x1, . . . ,xn] is a nonempty set which is closed under addition and satisfies hf ∈ I whenever h ∈ k[x1, . . . ,xn] and f ∈ I. We say that I is finitely generated if there exist f1, . . . , fs ∈ I such that every element of I is of the form h1f1+· · ·+hsfs for some h1, . . . ,hs ∈ k[x1, . . . ,xn]. By a fundamental result called the Hilbert Basis Theorem, every polynomial ideal is finitely generated. In fact, every polynomial ideal has a special finite generating set called a Gr¨obner basis. A Gr¨obner basis {g1, . . . , gt} is a set of polynomials in I such that the ideal generated by the leading terms of polynomials in I is equivalent to the ideal generated by the leading terms of g1, . . . , gt. Gr¨obner bases were established by Bruno Buchberger, and can be computed by the algorithm he developed in his 1965 PhD thesis. Over time, improvements to Buchberger’s Algorithm have been made in order to make the computation more efficient. Gr¨obner bases have many valuable applications in commutative algebra. By their nature, they provide a means of ideal description. They also give a definitive result in determining whether or not a polynomial is a member of an ideal. Gr¨obner bases are utilized in algebraic geometry as well, such as in using elimination theory to solve a system of polynomial equations, and in providing an implicit representation of a parametric curve or surface. Lastly, Gr¨obner bases can be used to describe the results of certain operations on ideals, such as the intersection of two ideals and the radical of an ideal, the latter being fundamental to the relationship between ideals and varieties.

Keywords

Abstract Algebra; Gr¨obner Bases; Algebraic Geometry; Communative Algebra

Document Type

Thesis

Year of Completion

2016

Advisor

Dr. Jozsef Losonczy

Academic Department

Mathematics

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