Real Plane Algebraic Curves
Permanent lenke
https://hdl.handle.net/10037/21691Dato
20210618Type
Master thesisMastergradsoppgave
Forfatter
González García, PedroSammendrag
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.
Forlag
UiT Norges arktiske universitetUiT The Arctic University of Norway
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