Newton polygons for L-functions of exponential sums of polynomials of degree six over finite fields Original Research Article

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)set membership, variantFq[x1,…,xn] is a polynomial, then one forms the exponential sum Sm( f)=∑(x1,…,xn)set membership, variant(Fqm)nΨ(TrFqm/Fp( f(x1,…,xn))). The corresponding L-function is defined by image. In this paper, we apply Dworkʹs 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)=6. As applications, we also give affirmative answers to Wanʹs conjecture and Hongʹs conjecture for the case deg f(x)=6.