Stretching Newton polygons using pure polynomials

TL;DR

This paper studies how the Newton polygon of a polynomial behaves under composition, showing it can be stretched horizontally when the inner polynomial has a pure Newton polygon, with implications for polynomial irreducibility.

Contribution

It demonstrates that composing with a polynomial having a pure Newton polygon stretches the Newton polygon of the outer polynomial, extending classical irreducibility results.

Findings

Newton polygon of composed polynomial is a horizontal stretch of the original

Pure Newton polygons lead to predictable changes under composition

Iterates of certain pure polynomials are irreducible

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 $f\circ g$ is the same as that of $f$, but stretched horizontally by $\operatorname{deg}(g)$. 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.