On plane curves given by separated polynomials and their automorphisms

TL;DR

This paper determines the automorphism groups of certain plane curves defined by separated polynomials over finite fields, providing new insights into their symmetries and applications in algebraic geometry codes.

Contribution

It computes the full automorphism group of specific separated polynomial curves and offers conditions characterizing their automorphisms, also analyzing the Norm-Trace curve case.

Findings

Automorphism group of the curve when m ≠ 1 mod p^n

Conditions for B(X) to have a single root in K

Automorphism group of the Norm-Trace curve

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 $p^n$ 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 $m \not\equiv 1 \pmod {p^n}$ 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 $\mathcal{C}: x^{(q^r-1)/(q-1)}=y^{q^{r-1}}+y^{q^{r-2}}+\ldots+y$ is computed. Finally, these results are used to construct multi point AG codes with many automorphisms.