Character sums over sparse elements of finite fields

We estimate mixed character sums of polynomial values over elements of a finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered basis over the subfield $\mathbb F_q$. First we use a combination of the inclusion-exclusion principle with bounds on character sums over linear subspaces to get nontrivial bounds for large $q$. Then we focus on the particular case $q=2$, which is more intricate. The bounds depend on certain natural restrictions. We also provide families of examples for which the conditions of our bounds are fulfilled. In particular, we completely classify all monomials as argument of the additive character for which our bound is applicable. Moreover, we also show that it is applicable for a large family of rational functions, which includes all reciprocal monomials.