Newton polygon of exponential sums in two variables with triangular base

Abstract

Let f(x1,x2) be a two-variable polynomial over a finite field of characteristic p, and assume that its convex hull is a triangle Δ. The goal of this paper is to study the Newton polygon NP(f,χ) of the L-function associated to f and a finite character χ of Zp. It is a partial generalization of [DWX], in which the authors studied polynomials of one variable.In this paper, we prove an improved lower bound IHP(Δ) for NP(f,χ), which, when f is non-ordinary, is strictly higher than the classical Hodge polygon. Moreover, we prove that if NP(f,χ) and IHP(Δ) coincide at one certain point, then they coincide at infinitely many points. This recovers the key property of the proof from [DWX].