Introduction

Abstract

In simplest terms, algebraic geometry is the study of the set of solutions of a system of polynomial equations. The main goal of real algebraic geometry is the study of real algebraic sets i.e. subsets of ℝ ⁿ defined by polynomial equations. By means of a simple example one can see some features which point up the difference between real and complex algebraic geometry. Let us consider the intersection of the straight line x = t, depending on the parameter t, with the cubic y ² = x ³ - x. For t = -1,0,1 the straight line is tangent to the cubic. In the complex plane, when t is different from -1,0,1, the straight line always intersects the cubic in two points. In the real plane the situation is more intricate.