Lower bound of overall exponential sums for polynomials \(\phi(x^ d)\) (Q1113951)

Let \(S^{(1)}(n,{\mathbb Q})\) denote the maximum module of exponential sums for polynomials of degree \(\leq n\) over the Galois field \(F_{\mathbb Q}\). In a previous paper the transition to the multiple exponential sums allowed us to obtain a good lower bound of the value \(S^{(1)}(n,\mathbb Q)\), which coincides with Weil's bound when \(n=q^{(m-1)/2}+1\); where \(q, m\) are odd and \(m\geq 3\). Here the same approach is used for the estimation of the value \(S^{(d)}(n,\mathbb Q)\), which corresponds to polynomials \(\phi (x^ d)\) over \(\mathbb F_{\mathbb Q}\), where \(d\) is any divisor of \(q-1\).