On plane curves given by separated polynomials and their automorphisms

Let ?"' be a plane curve defined over the algebraic closure K of a finite prime field ?"½p by a separated polynomial, that is ?"' : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of ?"' when m a‰¢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of ?"' to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve ?"' : X(qjavax.xml.bind.JAXBElement@42c3fbd6-1)/(q-1) = Yqjavax.xml.bind.JAXBElement@2638b9c6 + Yqjavax.xml.bind.JAXBElement@3cec48ce + ... + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.