论文标题
功率序列的非线性性能超过了阳性特征的完整非架构领域
Non-Linearizability of power series over complete non-Archimedean fields of positive characteristic
论文作者
论文摘要
在[H-Y83]中，Herman和Yoccoz证明，对于任何给定的本地分析（$ z = 0 $）功率系列$ f（z）= z（λ+\ sum_ {i = 1}^\ infty a_iz^i = 1}^\ infty a_iz^i）$在完整的非构造$ 0 $ 0 $ 0 $ = 1 $ = 1 $ = 1 $ = 1 $ 0 $ 0 $ 0 $ 0 $ 0的非构造范围内在$ z = 0 $。他们在积极特征的领域提出了同样的问题。在本文中，我们证明，相反，在这种情况下，大多数这样的功率系列更可能是不可分割的。更准确地说，给定一个正面特征的完整非Archimedean字段$ \ MATHCAL K $和功率系列$ f（z）= z（λ+\ sum_ {i = 1}^\ infty a_iz^i）\ in \ mathcal k [\！[z] \！ （标准〜\ star）对于$ f $不可接近可线化。最初，赫尔曼（Herman）在[her87，p 147]中猜想了在积极特征（不可连接）领域的电源序列的这种现象，并由lindahl作为[lin04，coptiune 2.2]提出了一个具体的问题。作为我们的标准的应用，我们证明了三个多项式家族的非可连接性。
In [H-Y83], Herman and Yoccoz prove that for any given locally analytic (at $z=0$) power series $f(z)=z(λ+\sum_{i=1}^\infty a_iz^i)$ over a complete non-Archimedean field of characteristic $0$ if $|λ|=1$ and $λ$ is not a root of unity, then $f$ is locally linearizable at $z=0$. They ask the same question for power series over fields of positive characteristic. In this paper, we prove that, on opposite, most such power series in this case are more likely to be non-linearizable. More precisely, given a complete non-Archimedean field $\mathcal K$ of positive characteristic and a power series $f(z)=z(λ+\sum_{i=1}^\infty a_iz^i) \in \mathcal K[\![z]\!]$ with $λ$ not a root of unity and $|1-λ|<1$, we prove a sufficient condition (Criterion~\star) for $f$ to be non-linearizable. This phenomenon of prevalence for power series over fields of positive characteristic being non-linearizable was initially conjectured in [Her87, p 147] by Herman, and formulated into a concrete question by Lindahl as [Lin04, Conjecture 2.2]. As applications of our criterion, we prove the non-linearizability of three families of polynomials.
