论文标题
有限场上的低度置换合理功能
Low-degree permutation rational functions over finite fields
论文作者
论文摘要
我们在f_q（x）中确定所有度量-4合理函数f（x），该功能在forute p^1（f_q）中确定，并回答了Ferraguti和Micheli的两个问题，内容涉及此类功能的数量以及此类功能的等价类别的数量，直至用学位为单位理性功能组成。我们还确定了f_q（x）中的所有程度-8有理函数f（x），如果q足够大，则在p^1（f_q）中取消了p^1（f_q），并且对于q是奇数或f（x）的情况，对第32度是相同的。此外，对于大多数其他正整数n <4096，对于每个足够大的Q，我们确定f_q（x）中的所有度n合理函数f（x），这些函数在f_q（x）中取消p^1（f_q），而不是f_q（x）中低度合理函数的组成。这些结果中的一些是通过在所有理性函数之间使用新的加洛伊斯理论表征（线性化）多项式的，这是独立关注的。
We determine all degree-4 rational functions f(X) in F_q(X) which permute P^1(F_q), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational functions. We also determine all degree-8 rational functions f(X) in F_q(X) which permute P^1(F_q) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for most other positive integers n<4096, for each sufficiently large q we determine all degree-n rational functions f(X) in F_q(X) which permute P^1(F_q) but which are not compositions of lower-degree rational functions in F_q(X). Some of these results are proved by using a new Galois-theoretic characterization of additive (linearized) polynomials among all rational functions, which is of independent interest.
