The number of rational points of a class of superelliptic curves

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