Real Plane Algebraic Curves

Abstract

This paper provides an elementary exposition of affine and projective real plane curves. Where possible, elementary proofs are given, but to some extent, one must make use of results from algebraic geometry (over the complex numbers) and real algebraic geometry. The bulk of the paper discusses algebraic aspects of affine and projective curves, but there is a short section at the end on topological aspects. A major theme of the paper is that real affine curves VR(f) with f an indefinite polynomial in R[X, Y ] have similar properties to complex affine curves, but semidefinite polynomials give rise to much different behavior. Many examples are given to illustrate the concepts.