Newton Polygon and Branching of Algebraic Curves

This Demonstration shows how to understand the branching type of an algebraic curve (near the origin, through which it is assumed to pass) using its Newton polygon. For a curve with an equation (in ) chosen from the popup menu, the graphic on the left shows the corresponding Newton diagram (red points) and the Newton polygon (lines connecting some of the points). The inverses of the slopes of the lines are the numbers such that near zero the curve looks like a union of curves , where is some constant. The values of are shown below the diagram. The actual curve in a neighborhood of the origin is shown to the right. The size of the neighborhood can be controlled using the sliders.


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The Newton diagram of a polynomial in two variables is the lattice of points in with coordinates . The lower convex hull of a Newton diagram is called its Newton polygon. Newton invented these polygons in his proof of what is called the Newton–Puiseux theorem [1]. The method has many applications, including the description of the branching type of algebraic curves demonstrated here.
[1] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge: Cambridge University Press, 2009.
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