Using Stepanov's method for exponential sums involving rational functions

For any additive character ψ and multiplicative character χ on a finite field Fq, and rational functions f,g in Fq(x), we show that the elementary Stepanov–Schmidt method can be used to obtain the corresponding Weil bound for the sum ∑x∈Fq⧹Sχ(g(x))ψ(f(x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values ωi of modulus q1/2 and the number of modulus 1.