## 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

Zbl 0621.10024
Full Text:

### References:

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