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Published online by Cambridge University Press: 17 March 2022
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.
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