Generic Newton polygons for $L$-functions of $(A,B)$-exponential sums

TL;DR

This paper proves that the Newton polygon associated with a class of $(A,B)$-exponential sums over finite fields is generically ordinary under certain conditions, confirming the Adolphson--Sperber conjecture for these cases.

Contribution

It establishes the generic ordinarity of Newton polygons for a broad class of $(A,B)$-polynomial exponential sums, verifying a key conjecture in the field.

Findings

Newton polyhedron $ riangle$ is generically ordinary if $p ot ot ext{divisible by } D$

Confirms the Adolphson--Sperber conjecture for these Newton polyhedra

Provides conditions on $p$ for ordinarity of the associated $L$-functions

Abstract

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 $p\equiv 1 \mod D$, where $D$ is a constant only determined by $\Delta$. In other words, we prove that the Adolphson--Sperber conjecture is true for $\Delta$.