有限线性递归数列子群的一些基本结果
Some basic results on finite linear recurring sequence subgroups
摘要：
一个$f$ -子群是一个线性递归序列子群，一个域的乘法子群，其元素可以通过线性递归关系生成（没有重复），具有特征多项式$f$。如果它可以以非循环方式生成（即，对于$f$的零$α$，不以$α^i, α^{i+1}, α^{i+2} \ldots$的顺序），则称为非标准，否则称为标准。我们将证明有限$f$ -子群必然由$f$的零点子集生成。利用这个结果改进了Brison和Nogueira的一个新定理。Brison和Nogueira提出的一个老问题是，是否存在自动生成的非标准$f$ -子组、$f$ -子组，这些子组不能由$f$的零生成。我们通过构建无限多的例子来肯定地回答这个问题。
英文摘要：
An $f$-subgroup is a linear recurring sequence subgroup, a multiplicative subgroup of a field whose elements can be generated (without repetition) by a linear recurrence relation, with characteristic polynomial $f$. It is called non-standard if it can be generated in a non-cyclic way (that is, not in the order $α^i, α^{i+1}, α^{i+2} \ldots$ for a zero $α$ of $f$), and standard otherwise. We will show that a finite $f$-subgroup is necessarily generated by a subset of the zeros of $f$. We use this result to improve on a recent theorem of Brison and Nogueira. A old question by Brison and Nogueira asks if there exist automatically non-standard $f$-subgroups, $f$-subgroups that cannot be generated by a zero of $f$. We answer that question affirmatively by constructing infinitely many examples.
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