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

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 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 $\mathbb F_{q^n}$

Parabolic curves for diffeomorphisms in C2

We give a simple proof of the existence of parabolic curves for diffeomorphisms in (C 2 , 0) tangent to the identity with isolated fixed point