Construção de bases de Gröbner para ideais polinomiais sobre um corpo K

Abstract

This work presents an introduction to the theory of Gröbner bases for polynomial ideals in the ring K[x1,...,xn] over an arbitrary field K. The aim of this work is to develop the main foundations of this theory, establishing the results necessary for the construction and characterization of Gröbner bases. To this end, it adopts a theoretical approach, in which the concepts, definitions, and proofs are developed progressively. Initially, it presents fundamental results on ring theory, monomial orderings, and the division algorithm in several indeterminates, extending to the multivariate case tools previously available only for polynomials in one indeterminate. Next, it characterizes monomial ideals and proves Dickson's Lemma and Hilbert's Basis Theorem, from which the existence of Gröbner bases for every polynomial ideal follows. Subsequently, it establishes Buchberger's Criterion and Buchberger's Algorithm, culminating in the characterization of minimal and reduced Gröbner bases. The results developed make it evident that Gröbner bases provide effective methods for solving problems involving polynomial ideals and can serve as a foundation for the future study of applications of the theory.