Stretching Newton polygons using pure polynomials

2024/05/17 by Gajek-Leonard, Rylan, Tomer, Uri
#11S82 #FOS: Mathematics #Number Theory (math.NT) #Primary 11S05 #Secondary 11R09

paper · doi:10.48550/arxiv.2405.10926

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 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.

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