论文标题
多项式在除法环上的固定点
Fixed Points of Polynomials over Division Rings
论文作者
论文摘要
我们研究了分区戒指$ d $的标准（或左）多项式$ f（x）$的离散动力。我们将它们的固定点定义为d $中的点$λ\ in \ f^{\ circ n}（λ）=λ$的任何$ n \ in \ mathbb {n} $，其中$ f^{\ circ n}（x）（x）$由$ f^{\ circ n} $ circ n}（x）（x）（x）（x）（x）（x）（x）（x）（x）（x）（ $ f^{\ circ 1}（x）= f（x）$。周期点类似地定义。我们证明，$λ$是$ f（x）$的固定点，并且仅当$ f（λ）=λ$的固定点可以使用多项式方程理论的已知结果得出结论，即任何$ M \ geq 2 $的多项式$ m \ geq 2 $最多都有$ m $ m $ M $ CONJUGACY CONJUGACY类别的固定点。我们还考虑了任意的周期性观点，并表明，通常它们的行为不像交换案例那样。我们提供了足够的条件，可以按预期进行周期性的行为。
We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings $D$. We define their fixed points to be the points $λ\in D$ for which $f^{\circ n}(λ)=λ$ for any $n \in \mathbb{N}$, where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$. Periodic points are similarly defined. We prove that $λ$ is a fixed point of $f(x)$ if and only if $f(λ)=λ$, which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most $m$ conjugacy classes of fixed points. We also consider arbitrary periodic points, and show that in general, they do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.
