Newton Polygon - Definition

Definition

A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots.

Let be a local field with discrete valuation and let

with . Then the Newton polygon of is defined to be the lower convex hull of the set of points

ignoring the points with . Restated geometrically, plot all of these points P_i on the xy-plane. Let's assume that the points indices increase from left to right (P₀ is the leftmost point, P_n is the rightmost point). Then, starting at P₀, draw a ray straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point P_k1 (not necessarily P₁). Break the ray here. Now draw a second ray from P_k1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point P_k2. Continue until the process reaches the point P_n; the resulting polygon (containing the points P₀, P_k1, P_k2, ..., P_km, P_n) is the Newton polygon.

Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P₀, ..., P_n. Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points.