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On a conjecture of Chalk. (English) Zbl 0876.11044

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 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.
Reviewer: J.Hinz (Marburg)

MSC:

11L07 Estimates on exponential sums

Citations:

Zbl 0621.10024
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References:

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