Grobner bases and their applications

Abstract

This thesis develops the fundamental theory of computational algebraic geometry with emphasis on polynomial ideals and Grobner bases. The preliminary part of the work is based on standard material from Ideals, Varieties, and Algorithms [1], and includes the study of polynomial ideals, affine varieties, monomial orderings, the division algorithm, Dickson’s Lemma, and the Hilbert Basis Theorem, which establish the algebraic and algorithmic foundations of the subject. The central theme of the thesis is Grobner bases and their computational properties. We present the theory of Grobner bases along with key algorithmic tools such as Buchberger’s algorithm and elimination theory. The role of Grobner bases in solving systems of polynomial equations is highlighted through applications of the Hilbert Nullstellensatz and methods for computing elimination ideals and minimal polynomials. In addition, the thesis explores polynomial mappings and their algebraic structure using Grobner basis techniques, illustrating how algebraic information can be extracted computationally from ideals. Finally, applications in combinatorial algebra are discussed, particularly in graph theory. Algebraic encodings of the k-coloring problem are used to study 3-colorable and general k-colorable graphs through associated polynomial ideals. These results demonstrate how Grobner basis methods provide a powerful bridge between algebraic geometry and combinatorial problems. Overall, the thesis emphasizes Grobner bases as an effective computational framework for translating geometric and combinatorial problems into algebraic form.