Stretching Newton polygons using pure polynomials

TL;DR

The paper investigates how the $p$-adic Newton polygon $\operatorname{NP}_p(f)$ behaves under composition with a $p^r$-pure polynomial $g$. The authors prove that, when $\operatorname{NP}_p(f)$ has $N$ segments with slopes $s_1<\cdots<s_N$ and $r+s_i>0$ for all $i$, the polygon of $f\circ g$ has the same number of segments with slopes $s_i/\deg g$ and with horizontal scaling by $\deg g$, effectively stretching the polygon. This yields that if $f$ is $p^r$-pure (or $p^r$-Dumas with $\gcd(r,\deg f)=1$), one can choose $g$ to preserve these properties, leading to dynamic irreducibility of iterates (recovering Odoni's classical results for Eisenstein polynomials). The results have concrete implications, including irreducibility of Taylor polynomials $f_n$ under composition with certain $p^r$-Dumas polynomials, thereby linking local Newton polygon geometry with arithmetic dynamics. Overall, the work provides a robust framework to deduce irreducibility of iterates from $p$-adic Newton polygon geometry and offers new avenues for studying dynamic irreducibility via 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 $f\circ g$ 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.

Figures (4)

• Figure 1: Newton polygons of $f$ and $f\circ g$ as in Theorem \ref{['main']}, where $d=\deg g$ and $i_j$ denote the $x$-coordinates of the vertices of $\operatorname{NP}_p(f)$.
• Figure 2: In bold are the lines $L_i$, which serve as lower bounds (in the sense of \ref{['lbL']}) for the Newton polygons of $a_ig^i$. The dashed line is the lower convex hull of all the $L_i$, which is exactly the Newton polygon of $f\circ g$.
• Figure :
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Theorems & Definitions (30)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• proof
• Remark 1.4
• Remark 2.1
• Example 2.2
• Theorem 2.3
• proof
• Lemma 2.4