# Newton Polygons

This page contains an applet created by Clif Presser and Rich Williams. The applet takes a given polynomial f(x) with integer coefficients and a given prime p and constructs the Newton polygon of f(x) with respect to p. The Newton polygon of f(x) with respect to p is defined as follows. If m is a non-zero integer and p is a prime, we define v(m) to be the largest non-negative integer r such that pr divides m. If the degree of f(x) is n and if the coefficient of xj is a non-zero integer aj, then we consider the point (j,v(aj)). The Newton polygon of f(x) with respect to p is the lower convex hull of the set of all such points. (More details about Newton polygons and their connection to p-adic roots of polynomials will be forthcoming.)

INSTRUCTIONS: Type a polynomial and a prime in the indicated spaces below. The polynomial should have integer coefficients. An example of an admissable form for the polynomial is:

2x^15 - 36*x^24 + 8x^10+56x^3-128
Observe that * can be used to indicate multiplication but this is not necessary. Also, spacing and ordering of the terms can vary. You can also type beyond the space allowed in the box below. After typing in a polynomial and a prime, simply click on the "Display" button. The Newton polygon should appear below.

## The Newton Polygon Applet

presser@sc.edu
williams@math.sc.edu
filaseta@math.sc.edu