An improved Mordell type bound for exponential sums

Abstract

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