Bounds for exponential sums over finite fields (Q1904528)

This paper is concerned with exponential sums defined over finite fields \(\mathbb{F}_{q^s}\), where \(q\) is a power of a prime. Such a sum involving a polynomial \(f(x)\) of one variable is transformed into a sum over \(\mathbb{F}_q\), by substituting \(x= x_1 b_1+ \dots+ x_s b_s\), where \(b_1, \dots, b_s\) is a vector space basis for \(\mathbb{F}_{q^s}\) over \(\mathbb{F}_q\). This produces a new polynomial in \(s\) variables, to which Deligne's bounds may be applied. In certain cases this leads to results superior to those following from Weil's estimate. The method is also applied to diagonal forms in more than one variable.