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Pure and mixed exponential sums. (English) Zbl 0937.11031
The authors consider bounds for the exponential sums \[ \text{(A)}\quad S(f,p^m)= \sum_{x=1}^{p^m} e_{p^m} (f(x)), \qquad \text{(B)}\quad S(\chi,f,p^m)= \sum _{\substack{ x=1,\\ p\nmid x}}^{p^m} \chi(x) e_{p^m} (f(x)), \] where \(p^m\) is a prime power with \(m\geq 2\), \(\chi\) is a multiplicative character modulo \(p^m\), \(e_{p^m}(x)= e^{2\pi ix/p^m}\) and \(f\) is a polynomial with integer coefficients and degree \(d>1\). For (A), the critical points are defined by \(p^{-t} f'(x)\equiv 0\bmod p\), where \(t= \operatorname {ord}_p (f')\). A consequence of the more precise estimates in the paper is \[ |S(f,p^m)|\leq (d-1) p^{t/(M+1)} p^{m(1-1/(M+1))}, \] providing \(m\geq t+2\), where \(M\) is the maximum multiplicity of the critical points. The estimates are obtained by a refinement of Chalk’s approach involving reduction of the sum locally around each critical point. Similar results have been obtained recently by P. Ding [J. Number Theory 65, 116-129 (1997; Zbl 0876.11044)] and W. K. A. Loh [Bull. Aust. Math. Soc. 50, 451-458 (1994; Zbl 0833.11040)]. Bounds for (B) follow by a similar technique by treating the sum as a twisted version of (A). The character \(\chi\) can be defined by \(\chi(a^k)= e(ck/p^{m-1} (p-1))\), where \(a\) is a fixed primitive root modulo \(p\) chosen so that \(a^{p-1}= 1+rp\) with \(p\nmid r\) and \(c\) is uniquely determined by \(\chi\) and \(a\) and \(0\leq c\leq p^{m-1} (p-1)\). Let \(t= \operatorname {ord}_p (f'(x))\), \(t_1= \operatorname {ord}_p (rxf'(x)+c)\). The critical points are defined by \(p^{-t_1} (rxf'(x)+c)\equiv 0\bmod p\). A consequence of much more precise and detailed estimates is \[ |S(\chi,f,p^m)|\leq dp^{t/(M+1)} p^{m(1-1/(M+1))}, \] providing \(m\geq t_1+2\), where \(M\) is the maximum multiplicity of the critical points.

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