ABSTRACT
This chapter provides an overview of some notions and results regarding the Zariski topology in a vector space over an algebraically closed field. It deals with algorithms for working with closed sets in an affine space. The main cornerstone of all these algorithms is the concept of Grobner bases. The chapter also provides an introduction of Grobner bases, the algorithm to compute them, and some applications that are particularly useful when dealing with algebraic groups. Grobner bases provide computational tools to solve various problems regarding ideals in polynomial rings. Their main drawback is that in practice they can be difficult to compute. The chapter describes the tangent space to a closed set at a given point. This of the highest importance for the study of algebraic groups, as the tangent space at the identity is the Lie algebra of such a group.
