A note on multiple exponential sums in function fields

Let A=Fq[t] denote the ring of polynomials over the finite field Fq. We denote by e a certain non-trivial character of the field of formal power series in terms of 1/t over Fq. For a monic g∈A and a polynomial G(x1,…,xd)∈A[x1,…,xd], we define S(G;g)=∑xe(G(x)/g) where x runs over (A/d(g)). In this paper, we prove that if G(x1,…,xd) is a homogeneous polynomial of degree k and if the g.c.d. of g and the coefficients of G is 1, then , where 〈g〉=qdegg and ϵ is a small positive constant.