Abstract: Let F_q be the finite field of q elements whose characteristic is p. Let F_q［t］ be the polynomial ring over F_q. Let e(·)denote a certain non-trivial character of the field of formal Laurent series in 1/t over F_q. Let k∈N with k≥2, a,b∈F_q［t］ and m=(m1,…,m_k)∈(F_q［t］)^k. Define the complete exponential sumS_k(a/b,m)=∑_{d∈Fq［t］, deg d<deg b}e(a/b∑^k_i=1m_id ⁱ).The following result is proved. Suppose that b≠0, gcd(b,a)=1 and gcd(b,m1,…,m_k)=1. If p>k, then |S_k(a/b,m)|≤C_k|b|1-1/k, where C2=1 and C_k=(k-1)2(k-1)⁽2k)/(k-2) when k≥3.

Key words: function field, exponential sum, Weyls differencing