EXPLICIT EVALUATION OF CERTAIN EXPONENTIAL SUMS OF QUADRATIC FUNCTIONS OVER Fpn, p ODD

x∈Fpn en(f(x)), where en(y) = e2πiTrn(y)/p, y ∈ Fpn , Trn = TrFpn/Fp . There is an effective way to compute the nullity of the quadratic form Trmn(f(x)) for all integer m > 0. Assuming that all such nullities are known, we find relative formulas for S(f,mn) in terms of S(f, n) when νp(m) ≤ min{νp(αi) : 1 ≤ i ≤ k}, where νp is the p-adic order. We also find an explicit formula for S(f, n) when ν2(α1) = · · · = ν2(αk) < ν2(n). These results generalize those by Carlitz and by Baumert and McEliece. Parallel results with p = 2 were obtained in a previous paper by the second author.