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On Hau's lemma

Published online by Cambridge University Press:  17 April 2009

W.K.A. Loh
Affiliation:
Department of MathematicsImperial CollegeHuxley Building 180 Queen'sGate London SW7 2BZUnited Kingdom
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Abstract

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Let f ∈ ℤ[X] and let q be a prime power pl(l ≥ 2). Hua stated and proved that

for some unspecified constant C > 0 depending on the derivative f of f; M denoting the maximum multiplicity of the roots of the congruence ptf(x) ≡ 0 (mod p), where t is an integer chosen so that the polynomial ptf(x) is primitive. An explicit value for C was given by Chalk for p ≥ 3. Subsequently, Ping Ding (in two successive articles) obtained better estimates for p ≥ 2.

This article provides a better result, based upon a more precise form of Hua's main lemma, previously overlooked.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Chalk, J.H.H., ‘On Hua's estimates for exponential sums’, Mathematika 34 (1987), 115123.CrossRefGoogle Scholar
[2]Ding, Ping, ‘An improvement to Chalk's estimation of exponential sums’, Acta Arith. LIX.2 (1991), 149155.CrossRefGoogle Scholar
[3]Hua, Loo-Keng, Additive theory of prime numbers (American Mathematical Society, Providence, 1965), pp. 27.Google Scholar
[4]Hua, Loo-Keng, ‘Die abschätzung von Exponentialsummen und ihre Anwendung in der Zahlentheorie’, Enzyklopädie der Math. Wiss Bd I2, H.13, TI (1959), p.41.Google Scholar
[5]Loxton, J.H. and Vaughan, R.C., ‘The estimation of complete exponential sums’, Canad. Math. Bull. 28 (1985), 440454.CrossRefGoogle Scholar