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On plane curves given by separated polynomials and their automorphisms (1708.05450v1)

Published 17 Aug 2017 in math.AG

Abstract: Let $\mathcal{C}$ be a plane curve defined over the algebraic closure $K$ of a prime finite field $\mathbb{F}p$ by a separated polynomial, that is $\mathcal{C}: A(y)=B(x)$, 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 well-known and studied in the literature. However just few informations are known on their automorphism groups. In this paper we compute the full automorphism group of $\mathcal{C}$ when $m \not\equiv 1 \pmod {pn}$ and $B(X)$ has just one root in $K$, that is $B(X)=b_m(X+b{m-1}/mb_m)m$ for some $b_m,b_{m-1} \in K$. Moreover, some sufficient conditions for the automorphism group of $\mathcal{C}$ to imply that $B(X)=b_m(X+b_{m-1}/mb_m)m$ are provided. As a byproduct, the full automorphism group of the Norm-Trace curve $\mathcal{C}: x{(qr-1)/(q-1)}=y{q{r-1}}+y{q{r-2}}+\ldots+y$ is computed. Finally, these results are used to construct multi point AG codes with many automorphisms.

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