On lower bounds for incomplete character sums over finite fields (Q1383495)

The author considers character sums \(\sum_{j=1}^N \chi (f(x_j))\), where \(\chi \) is a multiplicative character of exponent \(s>1\) of the finite field \(F_q\), \(f\) is a polynomial, and the \(x_j\) are arbitrary elements of the field. Generalizing a result of \textit{S. A. Stepanov} [Proc. Steklov Inst. Math. 143, 187-189 (1980); translation from Tr. Mat. Inst. Steklova 143, 175-177 (1977; Zbl 0431.10023)] concerning the prime field, he shows that this character sum is trivial for some set of \(s-1\) distinct \(s\)th power free polynomials of degrees bounded from above in terms of \(q\), \(N\), and \(s\). In addition, analogous results are obtained for additive characters.