On incomplete exponential sums over finite fields (Q2730843)

Let \(\mathbb F_q\) be a finite field of characteristic \(p > 2\) with \(q\) elements, \( \chi\) a multiplicative character of exponent \(r\) and \( \psi\) an additive character of \(\mathbb F_q\). Let \(S\) and \(T\) be subsets of \(\mathbb F_q\), and \(G\) a subgroup of the multiplicative group \(\mathbb F_q^{*}\) of \(\mathbb F_q\). The author considers incomplete character and exponential sums of the form NEWLINE\[NEWLINE U(f,g,S,T)= \sum_{x \in S} \sum_{y \in T} \chi(f(x)+g(y)), NEWLINE\]NEWLINE NEWLINE\[NEWLINE V(f,g,S,T)= \sum_{x \in S} \sum_{y \in T} \chi(f(x)+g(y)) \psi(x^{k}+y^{l}), NEWLINE\]NEWLINE where \(f\), \(g\) are polynomials over \(\mathbb F_q\) of degree \(m,n \geq 1\), respectively, and incomplete exponential sums of the form NEWLINE\[NEWLINE W(G)= \sum_{x \in G} \chi(x) \psi(x^{l}), NEWLINE\]NEWLINE NEWLINE\[NEWLINE W(f,g,G)={ \sum_{x \in G}}^{ \prime} \chi(f(x)) \psi(g(x)), NEWLINE\]NEWLINE where \(f \neq u^{r}\), \(g \neq v^{p}-v\) are rational functions over \(\mathbb F_q\), and the prime in the summation means that the poles of \(f\) and \(g\) are excluded. NEWLINENEWLINENEWLINEIf \( \chi\) and \( \psi\) are nontrivial characters, the author extends the arguments of \textit{F. R. K. Chung} [J. Number Theory 49, 95--106 (1994; Zbl 0807.11038)] and shows that NEWLINE\[NEWLINE |U(f,g,S,T)|\leq (mnq|S|\cdot |T|)^{1/2}(1-|S|/mq)^{1/2}(1-|T|/nq)^{1/2} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE |V(f,g,S,T)|\leq (mnq|S|\cdot|T|)^{1/2}(1+(|S|-1)/(q-1)m)^{1/2} (1+(|T|-1)/(q-1)n)^{1/2}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEIf \( \psi\) is not trivial and \( \chi\) is not trivial on \(G\), the author proves that NEWLINE\[NEWLINE|W(G)|\leq (q-1,|G|)q^{1/2} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE |W(f,g,G)|\leq c(f,g)q^{1/2}, NEWLINE\]NEWLINE where \(c(f,g)\) is a positive constant depending only of the spectrum of poles of \(f\), \(g\) and the set of zeros of \(f\) in \( \overline{\mathbb F}_q\).