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Using Stepanov’s method for exponential sums involving rational functions. (English) Zbl 1093.11058
Let $${\mathbb F}_q$$ be a finite field with characteristic $$p$$ and $$\psi$$ (respectively $$\chi$$) be an additive (resp. multiplicative) character on $${\mathbb F}_q$$. For rational functions $$f$$ and $$g$$ with coefficients in $${\mathbb F}_q$$, we put $$S(\psi,f;\chi,g):=\sum\chi(g(x))\psi(f(x))$$, where the sum is taken over the points $$x\in {\mathbb F}_q$$ that are not poles of $$f$$ or $$g$$. Using algebro-geometric tools and the Riemann Hypothesis for an appropriate $$L$$-function F. N. Castro and C. J. Moreno [Proc. Am. Math. Soc. 128, No. 9, 2529–2537 (2000; Zbl 0991.11065)] gave the following upper bound $$| S(\psi,f;\chi,g)| \leq {C \cdot q^{1/2}}$$, where $$C=\deg(f)_\infty+l+l'+l"-2$$, $$\deg(f)_\infty$$ being the degree of the divisor of poles of $$f$$, $$l$$ the number of distinct zeros and poles of $$g$$, $$l'$$ the number of distinct poles (including $$\infty$$) of $$f$$ and $$l"$$ the number of finite poles of $$f$$ which are also zeros or poles of $$g$$.
In this paper the authors use the elementary Stepanov-Schmidt method to give a proof of the above bound. Moreover, they determine precisely the number of characteristic values $$\omega_i$$ of modulus $$q^{1/2}$$ and the number of modulus value $$1$$. First they show, by adapting a standard argument, that the $$L$$-function is a polynomial and compute the coefficient of the leading term, in terms of Gauss sums of the type $$S(\psi,x;\chi,x)$$. In order to conclude the proof, they need to bound for an arbitrary finite extension $${\mathbb F}_{q^n}$$ the sum of all $$S({\psi\circ\text{tr}},f;\chi\circ N,g)$$ for all multiplicative characters $$\chi$$ and nontrivial additive characters $$\psi$$, which are been by expressing it in terms of the quantities $$s_n(f,b;g,a):=\#\{x\in{\mathbb F}_{q^n}, \text{tr}(f(x))= b,N(g(x))=a\}$$ and using the Stepanov’s method to make a precise estimate of $$s_n(f,b;g,a)$$.

##### MSC:
 11L07 Estimates on exponential sums 11L03 Trigonometric and exponential sums (general theory)
exponential sums
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##### References:
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