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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)
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