Abstract
A formula discovered by L. Carlitz in 1935 finds an interesting application in permutation rational functions of finite fields. It allows us to determine all rational functions of degree three that permute the projective line \(\mathbb {P}^{1}(\mathbb {F}_{q})\) over \(\mathbb {F}_{q}\), a result previously obtained by Ferraguti and Micheli through a different method. It also allows us to determine all rational functions of degree four that permute \(\mathbb {P}^{1}(\mathbb {F}_{q})\) under a certain condition. (A complete determination of all rational functions of degree four that permute \(\mathbb {P}^{1}(\mathbb {F}_{q})\) without any condition will appear in a separate forthcoming paper.)
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Hou, Xd. A power sum formula by Carlitz and its applications to permutation rational functions of finite fields. Cryptogr. Commun. 13, 681–694 (2021). https://doi.org/10.1007/s12095-021-00495-x
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DOI: https://doi.org/10.1007/s12095-021-00495-x