In this paper, we study the number of $\mathbb F_{ q^n}$-rational points on the affine curve $\mathcal{ X}_{ d,a,b}$ given by the equation $$ y^d=ax\text{ Tr}(x)+b,$$ where $\text{ Tr}$ denote the trace function from $\mathbb F_{ q^n}$ to $\mathbb F_{ q}$ and $d$ is a positive integer. In particular, we present bounds for the number of $\mathbb F_{ q}$-rational points on $\mathcal{ X}_{ d,a,b}$ and, for the cases where $d$ satisfies a natural condition, explicit formulas for the number of rational points are obtained. Particularly, a complete characterization is given for the case $d=2$. As a consequence of our results, we compute the number of elements $\alpha$ in $\mathbb F_{ q^n}$ such that $\alpha$ and $\text{ Tr}(\alpha)$ are quadratic residues in $\mathbb F_{ q^n}$.
Categories: math.NT