Stretching Newton Polygons Using Pure Polynomials
Abstract
The p-adic Newton polygon is a visual tool that encodes information about the roots and factorization of a polynomial relative to a prime p. In this article, we investigate how the Newton polygon changes under polynomial composition. If f and g are polynomials with rational (or p-adic) coefficients and the Newton polygon of g is pure (has only one segment), we show under some mild conditions that the Newton polygon of f°g is the same as that of f, but stretched horizontally by degg. When f=g, this implies that all iterates of certain pure polynomials are irreducible, recovering a classical result of Robert Odoni on the irreducibility of iterated Eisenstein polynomials.
Fields of Study & Subjects
Advanced Numerical Analysis Techniques
Robotic Mechanisms and Dynamics
Manufacturing Process and Optimization
Publication details
- Journal / source
- American Mathematical Monthly
- Publisher
- Informa UK Limited
- Volume
- 133
- Issue
- 6
- Pages
- 532–539
- Publication date
- 15 April 2026
- Language
- EN