Newton-Puiseux algorithm

In this thesis, an explanation of the Newton-Peiseux algorithm is given. This explanation is supplemented with well-worked and explained examples of how to use the algorithm to find fractional power series expansions for all branches of a polynomial at the origin. The expansions that can be obtained by the algorithm are then used to classify the local behavior of 2nd, 3rd, and 4th degree polynomials at a point. This is done by enumerating the Newton polygons for each degree and by careful examination of the properties of each of the Newton polygons.