仿射空间自同构的驯服群的无限传递性
Infinite transitivity of tame groups of automorphisms of affine spaces
AI总结：
该研究针对有限域上n维仿射空间的驯服自同构群，探究其对m个点有序子集的作用，构造可映射有序序列的驯服自同构并证明其复杂度的上下界，运用多数学分支方法获得相关定性与定量结果。
AI中文摘要：
对于正整数n和m，我们研究有限域上n维仿射空间的驯服自同构群对m个点的有序子集的作用。我们的主要兴趣在于构造将一个有序序列映射到另一个有序序列的驯服自同构，并证明这类自同构的最大复杂度的上下界。我们偏好的自同构复杂度度量是定义它及其逆的多项式的次数的最大值。利用来自数学多个分支的方法与结果，包括对称群理论、有限域上的仿射几何、多项式插值、域上射影空间的组合学以及多项式自同构，我们获得了大量定性和定量结果。
英文摘要：
For positive integers $n$ and $m$, we study the actions of the groups of tame automorphisms of the $n$-dimensional affine spaces over finite fields on ordered subsets of $m$ points. Our primary interest lies in constructing tame automorphisms that take one ordered sequence to another and in proving upper and lower bounds on the maximal complexity of such automorphisms. Our preferred measure of complexity of an automorphism is the maximum of the degrees of the polynomials that define it and its inverse. Using methods and results from various branches of mathematics, including the theory of symmetric groups, affine geometry over finite fields, polynomial interpolation, combinatorics of projective spaces over fields, and polynomial automorphisms, we obtain a wide variety of qualitative and quantitative results.
