Real Plane Algebraic Curves

Abstract

This master thesis studies several properties of real plane algebraic curves, focusing on the case of even degree. The question of the relative positions of the connected components of real plane algebraic curves originates in Hilbert's sixteenth problem which, despite its prominence, is still open in the case of higher degree curves. The goal of this thesis is an exposition of fundamental contributions to this problem, which have been obtained within the last century. The main aim of the thesis is to clarify these and to make them more accessible. Chapter 1 gives a brief introduction into the study of real plane algebraic curves. The exposition of this chapter builds on the standard knowledge which are normally obtained in an undergraduate course of algebraic curves, which usually focus only on complex plane algebraic curves. In Chapter 2, several topological properties of real plane curves are developed. The main statements here can be mostly established from Bezout's theorem and its consequences. The main result presented in this chapter is Harnack's inequality and the classi cation of the curves until degree ve. The goal of Chapter 3 is to prove Petrowski's inequalities using Morse theoretic results along with the original arguments which appeared in Petrowski's manuscript. Chapter 4 presents results arising from the complexi cation of a real plane curve. Finally, Chapter 5 mainly presents results from Smith theory. In particular, this allows to see how Smith's inequality generalizes Harnack's inequality which were presented in Chapter 2 to higher dimensions.