Explicit determination of a class of permutation rational functions in any characteristic
For researchers in finite fields and permutation polynomials, this provides a unified geometric framework that extends known results, though the contribution is incremental.
The authors explicitly determine a broad class of permutation rational functions of small degree that permute the multiplicative subgroup μ_{q+1}, and as an application, they explicitly determine many permutation quadrinomials over F_{q^2} induced by degree-3 rational functions. The work unifies and extends several existing results.
In this paper, we make use of the classification results of low-degree permutation rational functions together with their geometric properties to investigate rational functions that induce permutations on the multiplicative subgroup mu_q+1, where q is a prime power. By carefully analyzing the structural conditions under which such rational functions permute muq+1, we obtain an explicit description of a broad class of permutation rational functions of small degree. As a direct application of these findings, we explicitly determine many permutation quadrinomials over Fq2 that are induced by degree-3 rational functions permuting muq+1. Our approach not only unifies and extends several existing results in the literature but also provides a concrete geometric perspective for characterizing permutation polynomials over Fq2.