On upper bounds of Chalk and Hua for exponential sums
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- by Todd Cochrane and Zhiyong Zheng PDF
- Proc. Amer. Math. Soc. 129 (2001), 2505-2516 Request permission
Abstract:
Let $f$ be a polynomial of degree $d$ with integer coefficients, $p$ any prime, $m$ any positive integer and $S(f,p^m)$ the exponential sum $S(f,p^m)= \sum _{x=1}^{p^m} e_{p^m}(f(x))$. We establish that if $f$ is nonconstant when read $\pmod p$, then $|S(f,p^m)|\le 4.41 p^{m(1-\frac 1d)}$. 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 $S(f,p^m)$ with $x$ restricted to values congruent to $\alpha \pmod {p^m}$. We obtain $|S_\alpha (f,p^m)| \le \min \{\nu ,3.06\} p^{\frac t{\nu +1}}p^{m(1-\frac 1{\nu +1})}$ for $p$ 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 $|S_\alpha (f,p^m)| \le p^{m(1-\frac 1{\nu +1})}$.References
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Additional Information
- Todd Cochrane
- Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
- MR Author ID: 227122
- Email: cochrane@math.ksu.edu
- Zhiyong Zheng
- Affiliation: Department of Mathematics, Tsinghua University, Beijing, People’s Republic of China
- Email: zzheng@math.tsinghua.edu.cn
- Received by editor(s): June 3, 1999
- Published electronically: April 17, 2001
- Additional Notes: The research of the second author was supported by the National Science Fund of The People’s Republic of China for Distinguished Young Scholars.
- Communicated by: Dennis A. Hejhal
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2505-2516
- MSC (1991): Primary 11L07, 11L03
- DOI: https://doi.org/10.1090/S0002-9939-01-06189-5
- MathSciNet review: 1838371