An estimate for incomplete mixed character sums and applications

Let $q$ be a prime power and $m>1$ be any integer. Let ${\mathbb{F}}_{q^m}$ be the finite field of order $q^m$ and $\theta \in {\mathbb{F}}_{q^m}$ be such that ${\mathbb{F}}_{q^m}=\mathbb{F}(\theta )$. We obtain a nontrivial bound for the mixed character sum ${\sum }_{x\in \mathbb{F}}\chi (\theta +x)\psi (x)$, where $\chi$ and $\psi$ are multiplicative and additive characters of ${\mathbb{F}}_{q^m}$ and $\mathbb{F}$, respectively, using function field methods. As an application of our main result, we prove that for fixed $m$ and sufficiently large prime powers $q$, that satisfy certain conditions, ${\mathbb{F}}_{q^m}\mathrm{/}\mathbb{F}$ possesses the weak line property for primitive normal elements. In particular, our result is a strengthening of existing results.