Galois points and rational functions with small value sets (Q6545071)

The author establishes a connection between Galois points of plane curves and rational functions that exhibit small value sets. The primary theorem states that the defining polynomial of any plane curve admitting two Galois points is an irreducible factor of a polynomial obtained from the equality of two rational functions. This theorem is further supported by the assumption that the Galois groups of the two points generate their semidirect product.\N\NA notable result referenced in this paper is from Bartoli, Borges, and Quoos, which suggests that one of these rational functions over a finite field has a very small value set. The paper demonstrates that when two Galois points are external, the defining polynomial is an irreducible factor of the difference of two polynomials in one variable.\N\NThe paper is well-written and presents its arguments in a clear and structured manner. The connection between Galois points and rational functions with small value sets is novel and adds valuable insights to the existing literature on algebraic curves over finite fields. The use of polynomials and rational functions to describe these relationships is both innovative and effective.\N\NThe proofs provided are rigorous and well-supported by recent results in the field. The inclusion of multiple theorems and corollaries helps to solidify the paper's claims and provides a comprehensive understanding of the subject matter.