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

12/02/2020

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by   Daniela Oliveira, et al.

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Let 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 · P(x) - λ, where P(x) is a 𝔽_q-linearized polynomial and λ∈𝔽_q. Our main results provide a characterization of the number of rational points in some extension 𝔽_q^r of 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.