On a conjecture of Chalk (Q1360715)

Let \(f \in \mathbb{Z}[x]\) be a polynomial of degree \(g \geq 2\). Consider, for any positive integer \(q\), the complete trigonometric sum \[ S(q,f) = \sum_{k=0}^{q-1}\exp (2 \pi i f(k)/q). \] In 1987 \textit{J. H. H. Chalk} [Mathematika 34, 115-123 (1987; Zbl 0621.10024)] made a conjecture on an upper bound for \(S(q,f)\), when \(q\) is a prime power \(p^n\). In the present paper the author proves this conjecture if \(p\) is relatively small but \(\geq\) 3. When \(p \geq 3\) is relatively large, he gives a counterexample to Chalk's conjecture and derives an alternative upper bound which is best possible. Moreover, for \(p=2\), the author improves previous results.