On a conjecture of Wan about limiting Newton polygons

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 $f\mathrm{mod}\mathfrak{p}$ 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.