On upper bounds of Chalk and Hua for exponential sums期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On upper bounds of Chalk and Hua for exponential sums

Authors:	Todd Cochrane Zhiyong Zheng

Institution:	Department of Mathematics, Kansas State University, Manhattan, Kansas 66506 ; Department of Mathematics, Tsinghua University, Beijing, People's Republic of China

Abstract:	Let be a polynomial of degree with integer coefficients, any prime, any positive integer and the exponential sum $S(f,p^m)= \sum_{x=1}^{p^m} e_{p^m}(f(x))$ . We establish that if is nonconstant when read $\pmod p$ , then $\vert S(f,p^m)\vert\le 4.41 p^{m(1-\frac 1d)}$ . Let $t=\text{ord}_p(f')$ , let $\alpha$ be a zero of the congruence $p^{-t}f'(x) \equiv 0 \pmod p$ of multiplicity $\nu$ and let $S_\alpha(f,p^m)$ be the sum with restricted to values congruent to $\alpha \pmod {p^m}$ . We obtain $\vert S_\alpha (f,p^m)\vert \le \min \{\nu,3.06\} p^{\frac t{\nu+1}}p^{m(1-\frac 1{\nu+1})}$ for odd, $m \ge t+2$ and $d_p(f)\ge 1$ . If, in addition, $p \ge (d-1)^{(2d)/(d-2)}$ , then we obtain the sharp upper bound $\vert S_\alpha(f,p^m)\vert \le p^{m(1-\frac 1{\nu+1})}$ .

Keywords:	Exponential sums

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