On circulant matrices and rational points of Artin Schreier's curves

Let $\mathbb F_q$ be a finite field with $q$ elements, where $q$ is an odd prime power. In this paper we associated circulant matrices and quadratic forms with curves of Artin-Schreier $y^q - y = x \cdot P(x) - \lambda,$ where $P(x)$ is a $\mathbb F_q$-linearized polynomial and $\lambda \in \mathbb F_q$. Our main results provide a characterization of the number of rational points in some extension $\mathbb F_{q^r}$ of $\mathbb F_q$. In the particular case, in the case when $P(x) = x^{q^i}-x$ we given a full description of the number of rational points in term of Legendre symbol and quadratic characters.