The Newton polygon and the Puiseux characteristic

Abstract

In this thesis, we will use the Newton polygon and the Puiseux characteristic to study complex analytic curves in C{x,y} and C[[x,y]]. This allows us to topologically classify the plane curve singularities. Chapter 1 will introduce the Newton polygon, the process of sliding towards a root and polar curve. The first section of chapter 2 contains the technical background to this topic. The second section introduces the Puiseux characteristic, and the third uses results from knot theory to classify the plane curve singularities as the cone over an iterated torus knot. In the third chapter, we will look at the Kuo-Lu theorem, which is a generalisation of Rolle’s theorem to complex curves. Finally, in the fourth chapter, we will give an application of the previous results to show a method of calculating the Lojasiewicz exponent.