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.)

L. Carlitz 在 1935 年发现的一个公式在有限域的置换有理函数中找到了一个有趣的应用。它允许我们确定所有三次有理函数，它们置换投影线\(\mathbb {P}^{1}(\mathbb {F}_{q})\)在\(\mathbb {F}_{ q}\)，这是 Ferraguti 和 Micheli 之前通过不同方法获得的结果。它还允许我们确定在特定条件下置换\(\mathbb {P}^{1}(\mathbb {F}_{q})\) 的所有四次有理函数。（对所有四次有理函数的完全确定，这些函数在没有任何条件的情况下置换\(\mathbb {P}^{1}(\mathbb {F}_{q})\)将出现在单独的即将发表的论文中。）