On upper bounds of Chalk and Hua for exponential sums

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 $|S_\alpha (f,p^m)| \le p^{m(1-\frac 1{\nu +1})}$.