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Register a new userOn a conjecture of Wan about limiting Newton polygons
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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 $fmodmathfrak p$ 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.
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In this paper, we consider the following $(A, B)$-polynomial $f$ over finite field: $$f(x_0,x_1,cdots,x_n)=x_0^Ah(x_1,cdots,x_n)+g(x_1,cdots,x_n)+P_B(1/x_0),$$ where $h$ is a Deligne polynomial of degree $d$, $g$ is an arbitrary polynomial of degree $< dB/(A+B)$ and $P_B(y)$ is a one-variable polynomial of degree $le B$. Let $Delta$ be the Newton polyhedron of $f$ at infinity. We show that $Delta$ is generically ordinary if $pequiv 1 mod D$, where $D$ is a constant only determined by $Delta$. In other words, we prove that the Adolphson--Sperber conjecture is true for $Delta$.
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