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On a Conjecture of Chalk

✍ Scribed by Ping Ding

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 261 KB
Volume: 65
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1997.2125

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✦ Synopsis

Let f # Z[x] with degree k and let p be a prime. By a complete trigonometric sum we mean a sum of the form S(q, f )= q x=1 e q ( f (x)), where q is a positive integer and e q (:)=exp(2?if (x)Âq). Professor Chalk made a conjecture on the upper bound of S(q, f ) when q is a prime power. We prove Chalk's conjecture, in the affirmative, if p is relatively small but 3. When p 3 is relatively large, we give an alternative upper bound which is best possible. For p=2, we also improve previous results.

1997 Academic Press

Note that the inequality p t k is a trivial consequence of (3). Let r=r( f ) denote the number of distinct roots of the congruence p &t f $(x)#0(mod p) (0 x<p).

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