论文标题
分类出色的散射多项式
Towards the classification of exceptional scattered polynomials
论文作者
论文摘要
在过去几年中，有限场上的分散多项式吸引了人们越来越多的关注。原因之一是它们与最大秩距离（MRD）代码的深入联系。出色的分散多项式的已知分类结果，即散布在无限田间扩展上的多项式，仅限于其索引$ \ ell $小的索引$ \ ell $小，或者比$ q $ $ q $ $ k $ k $大的质量数字大，或者比$ k $在$ k $ k $ k $的情况下要小。在本文中，当$ \ ell $和$ k $之间的最大值奇怪时，我们完全对出色的散射多项式进行了分类，并在2021年的Ferraguti和Micheli的结果时给出部分结果。
Scattered polynomials over finite fields attracted an increasing attention in the last years. One of the reasons is their deep connection with Maximum Rank Distance (MRD) codes. Known classification results for exceptional scattered polynomials, i.e. polynomials which are scattered over infinite field extensions, are limited to the cases where their index $\ell$ is small, or a prime number larger than the $q$-degree $k$ of the polynomial, or an integer smaller than the $k$ in the case where $k$ is a prime. In this paper we completely classify exceptional scattered polynomials when the maximum between $\ell$ and $k$ is odd, and give partial results when it is even, extending a result of Ferraguti and Micheli in 2021.
