Binomial exponential sums

Abstract

We obtain new bounds of exponential sums modulo a prime $p$ with binomials $a{x}^k+b{x}^n$. In particular, for $k=1$, we improve the bound of Karatsuba (1967) from $O(n^{1/4}{p}^{3/4})$ to $O\left(p^{3/4}+n^{1/3}{p}^{2/3}\right)$ for any $n$, and then use it to improve the bound of Akulinichev (1965) from $O(p^{5/6})$ to $O(p^{4/5})$ for $n|(p-1)$. The result is based on a new bound on the number of solutions and of degrees of irreducible components of certain equations over finite fields.