On the number of rational points of Artin–Schreier’s curves and hypersurfaces

Let \(\mathbb {F}_{q^n}\) ${\mathbb{F}}_{q^n}$ represent the finite field with \(q^n\) $q^n$ elements. In this paper, our focus is on determining the number of \(\mathbb {F}_{q^n}\) ${\mathbb{F}}_{q^n}$ -rational points for two specific objects: an affine Artin–Schreier curve given by the equation \(y^q-y = x(x^{q^i}-x)-\lambda \) $y^q-y=x(x^{q^i}-x)-\lambda$ , and an Artin–Schreier hypersurface given by the equation \(y^q-y=\sum _{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-\lambda \) $y^q-y={\sum }_{j=1}^r{a}_j{x}_j(x_j^{q^{i_j}}-x_j)-\lambda$ . Additionally, we establish that the Weil bound is only achieved in these cases when the trace of the element \(\lambda \in \mathbb {F}_{q^n}\) $\lambda \in {\mathbb{F}}_{q^n}$ over the subfield \(\mathbb {F}_q\) ${\mathbb{F}}_q$ is equal to zero.