Galois points for double-Frobenius nonclassical curves

TL;DR

This paper investigates the distribution of Galois points on a special class of plane curves over finite fields, including the Dickson-Guralnick-Zieve curve, revealing new properties and solving an open problem in the field.

Contribution

It determines the distribution of Galois points for Frobenius nonclassical curves over finite fields, extending understanding of their geometric and algebraic properties.

Findings

Distribution of Galois points characterized for these curves

Includes analysis of the Dickson-Guralnick-Zieve curve

Addresses and modifies a previous open problem in Galois points theory

Abstract

We determine the distribution of Galois points for plane curves over a finite field of $q$ elements, which are Frobenius nonclassical for different powers of $q$. This family is an important class of plane curves with many remarkable properties. It contains the Dickson-Guralnick-Zieve curve, which has been recently studied by Giulietti, Korchmaros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.