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.

中文翻译：

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有限域稀疏元素上的字符和

我们估计有限域元素上多项式值的混合字符和${\mathbb{F}}_{q^r}$在子域上具有固定有序基础的稀疏表示 ${\mathbb{F}}_q$。首先，我们将包含-排除原理与线性子空间上的字符和的界限相结合，以获得大的非平凡界限$q$。然后我们重点关注具体案例$q=2$，这更加复杂。该界限取决于某些自然限制。我们还提供了满足我们界限条件的示例系列。特别是，我们将所有单项式完全分类为我们的界限适用的加性特征的参数。此外，我们还证明它适用于一大类有理函数，其中包括所有倒数单项式。