This chapter consists of basic real algebraic geometry. In Chapter 2 we studied basic properties of curves in the plane. We now generalize these properties to arbitrary varieties in any real space and introduce additional properties.

We begin the chapter by briefly discussing polynomial ideals. We then study the dimension of a real variety, connected components, irreducible components, tangent spaces, singular points, regular points, and other properties. We discuss why there is no one well-behaved definition for the degree of a real variety. We also study the projection of real varieties and partitioning polynomials in higher dimensions.

After seeing some basics of incidence theory, we wish to discuss how polynomial methods are used to study incidences. For that, we first need a basic introduction to algebraic geometry over the reals. In this chapter we focus mainly on the plane, postponing the treatment of higher dimensions to Chapter 4. This allows us to discuss several planar results in Chapter 3, before dealing with more involved algebraic geometry.

We begin the chapter by introducing varieties and their properties. We then focus on curves in the plane and their properties: degree, irreducible components, connected components, intersections, and more. We conclude the chapter with a polynomial proof of Pascal’s theorem, due to Plücker.