  We obtain new bounds of exponential sums modulo a prime $p$ with binomials
$ax^k + bx^n$. 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.
