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Exponential sums with rational function entries. (English) Zbl 0956.11018

In a previous paper [Acta Arith. 91, 249-278 (1999; Zbl 0937.11031)], the authors considered complete exponential sums modulo a prime power \(p^m\) with \(m\geq 2\) involving a polynomial with integer coefficients, and the sum was possibly twisted with a Dirichlet character \(\pmod{p^m}\) (accordingly, it was called ”pure” or ”mixed”). In the present paper, polynomials are replaced, more generally, by rational functions \(f(x)=f_1(x)/f _2(x)\). Up to special cases such as Kloosterman sums, exponential sums like this have been studied relatively little for higher prime powers. The estimates obtained depend on the total degree \(d(f)=d(f_1) + d(f_2)\) and on the maximal degree \(d^*(f)=\max (d(f_1), d(f_2))\) of \(f\), with \(d\) denoting the degree of a polynomial. Also, let \(d_p (f)\) and \(d_p ^*(f)\) have the same meaning if \(f\) is reduced \(\pmod{p}\).
As an example of the estimates for pure exponential sums, we may mention the bound \(dp^{m(1-\frac 1{d^*})}\) (Corollary 3.2) under the assumptions that \(p\) be an odd prime and \(d_p(f) \geq 1\). For \(p=2\), the same holds with an extra factor \(\sqrt{2}\). Further, putting \(D=\max (d(f), 2d(f_2))\), we have the estimate \(4Dp^{m(1-1/(D+1))}\) (Corollary 4.1) for a mixed sum, for all primes, if again \(d_p (f)\geq 1\).

MSC:

11L03 Trigonometric and exponential sums (general theory)
11L07 Estimates on exponential sums

Citations:

Zbl 0937.11031
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