On a conjecture of Wan about limiting Newton polygons

TL;DR

This paper proves that for certain polynomials over number fields, the Newton polygons do not converge at all primes, confirming the 'only if' part of Wan's conjecture in the rational case.

Contribution

It establishes the non-convergence of Newton polygons for polynomials with a global permutation polynomial factor, confirming a key part of Wan's conjecture over the rationals.

Findings

Newton polygons do not converge for these polynomials at all primes.

The result confirms the 'only if' part of Wan's conjecture in the rational case.

Abstract

We show that for a monic polynomial $f(x)$ over a number field $K$ containing a global permutation polynomial of degree $>1$ as its composition factor, the Newton Polygon of $f\mod\mathfrak p$ does not converge for $\mathfrak p$ passing through all finite places of $K$. In the rational number field case, our result is the "only if" part of a conjecture of Wan about limiting Newton polygons.