Bases de Gröbner e os Teoremas de Hilbert

Abstract

In this work, we will present Gröbner Bases, which are sets of polynomials that simplify the resolution of systems of polynomial equations by establishing an ordering among polynomials in K[x_1,...,x_n]. We will also discuss the Hilbert's Nullstellensatz, which is a fundamental idea in algebraic geometry that relates ideals in K[x_1,...,x_n] to affine varieties, showing that each variety has an associated ideal and vice versa. These concepts have a significant impact on mathematics, enabling the translation of geometric problems into algebraic ones and establishing a deep connection between algebra and geometry.