Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Abstract

Let Fq be the finite field of q elements with characteristic p and Fqm its extension of degree m. Fix a nontrivial additive character Ψ of Fp. If f(x1,…, xn)∈Fq[x1,…, xn] is a polynomial, then one forms the exponential sum Sm(f)=∑(x1,…,xn)∈(Fqm)nΨ(TrFqm/Fp(f(x1,…,xn))). The corresponding L functions are defined by L(f, t)=exp(∑∞m=0Sm(f)tm/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.