## 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

### Keywords:

exponential sums; character sums

Zbl 0937.11031
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