An improved Mordell type bound for exponential sums

Abstract:

For a sparse polynomial $f(x)=\sum _{i=1}^r a_ix^{k_i}\in \mathbb Z [x]$, with $p\nmid a_i$ and $1\leq k_1<\cdots <k_r<p-1$, we show that \[ \left |\sum _{x=1}^{p-1} e^{2\pi i f(x)/p} \right | \leq 2^{\frac {2}{r}} (k_1\cdots k_r)^{\frac {1}{r^2}}p^{1-\frac {1}{2r}}, \] thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.