Affine Algebraic Varieties

In this chapter we introduce the concepts of affine algebraic varieties and their associated ideals, which consist of polynomials that vanish on these varieties. In addition, we explore the properties of these varieties and ideals by using the theory of ideals developed so far. Furthermore, we provide a proof of the Hilbert Nullstellensatz, which establishes a fundamental relation between geometry and algebra. We then apply this theorem, along with the Gr¨obner basis theory, to analyze and study solutions of systems of polynomial equations.