Behaviour of Newton Polygon over polynomial composition

TL;DR

This paper investigates how Newton polygons behave under polynomial composition over the rationals, providing explicit descriptions of their structure and implications for dynamical irreducibility and number field stability.

Contribution

It establishes conditions for explicitly describing Newton polygons of composed polynomials, advancing understanding of their evolution under iteration.

Findings

Explicit description of Newton polygon vertices for compositions

Conditions linking Newton polygons of g(f^n(x)) and g(x)

Applications to dynamical irreducibility and number field towers

Abstract

In this paper, we study the structure of Newton polygons for compositions of polynomials over the rationals. We establish sufficient conditions under which the successive vertices of the Newton polygon of the composition $ g(f^n(x)) $ with respect to a prime $ p $ can be explicitly described in terms of the Newton polygon of the polynomial $ g(x) $. Our results provide deeper insights into how the Newton polygon of a polynomial evolves under iteration and composition, with applications to the study of dynamical irreducibility, eventual stability, non-monogenity of tower of number fields, etc.