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The estimation of complete exponential sums. (English) Zbl 0575.10033

Let f(X) be an integer polynomial of degree n and let \(S=S(f,q)=\sum^{q}_{x=1}\exp (2\pi if(x)/q)\). Estimates for S are of great importance in analytic number theory, in connection with Waring’s problem, for example. There are multiplicative formulae for S, so it suffices to consider the case \(q=p^{\alpha}\). Moreover one may easily reduce to the case in which p does not divide the content of f(X)-f(O). If \(\alpha =1\) one then has \(| S| \leq (n-1)p^{1/2}\) by Weil’s ”Riemann hypothesis” for curves over finite fields. For larger values of \(\alpha\) the situation is more complicated. Let e denote the largest multiplicity of any complex zero of f’(X). In the present paper a considerable advance is made by showing that \[ (*)\quad | S(f,p^{\alpha})| \leq C_ n p^{\alpha e/(e+1)}, \] with an explicit constant \(C_ n\). Previous bounds had \(e/(e+1)\) replaced by 1- 1/n [L.-K. Hua, J. Chin. Math. Soc. 2, 301-312 (1940; Zbl 0061.066)] and 1-1/(2e) [the first author and R. A. Smith, J. Lond. Math. Soc., II. Ser. 26, 15-20 (1982; Zbl 0474.10030)]. The constant \(C_ n\) depends on the divisibility by p of a certain ”different” of f’(X).
The proof is ”elementary”, but distinctly more involved than previous work in this area, a succession of p-adic approximations to the roots of f’(X) being used. The paper ends with a number of examples in which the estimate (*) is essentially best possible.
Reviewer: D.R.Heath-Brown

MSC:

11L40 Estimates on character sums
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