论文标题
确定一类置换四元素
Determination of a class of permutation quadrinomials
论文作者
论文摘要
我们确定表格x^r a（x^{q-1}）上f_ {q^2}上的所有置换多项式，其中，对于某些q，这是f_q的特征的Q+1（mod q+1）的特征的幂，并且所有条款的所有项都在{0，x）中具有{0，1，q，q，q+1}。然后，我们使用此分类来解决文献中的八个猜想和开放问题，我们表明，结果中最简单的特殊案例暗示了58种文献的结果。我们的证明是在以前似乎不适用的情况下对几何技术的新颖使用，即了解在小有限域中高度合理功能的算术，尽管在这种情况下，Weil界限并未提供有用的信息。
We determine all permutation polynomials over F_{q^2} of the form X^r A(X^{q-1}) where, for some Q which is a power of the characteristic of F_q, the integer r is congruent to Q+1 (mod q+1) and all terms of A(X) have degrees in {0, 1, Q, Q+1}. We then use this classification to resolve eight conjectures and open problems from the literature, and we show that the simplest special cases of our result imply 58 recent results from the literature. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high-degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.
