自动机与Nottingham群中的有穷级元
Automata and finite order elements in the Nottingham group
摘要：
在2处的诺丁汉群是在变量$t$中的（形式的）幂级数$t+a_2 t^2+ a_3 t^3+ \cdots$的群，其中系数$a_i$来自具有两个元素的域，其中群运算由幂级数的合成给出。这样一个系列的深度是最大的$d\geq 1$为其中$a_2=\dots=a_d=0$。只有极少数的有穷级的幂级数通过其系数的公式被明确地知道。我们认为，在本文中，它是有利的，通过自动机描述这样的系列在封闭的计算形式，有效版本的证明的Christol定理识别代数和自动系列的基础上。直到共轭为止，具有固定中断序列（即$σ^{\circ 2^i}$的深度序列）的$2^n$阶的序列只有200多个。本文从Witt向量或Carlitz模的构造出发，给出了以下问题的自动机理论描述：（a）具有间断序列（1，m）（m<10）的所有四阶级数的共轭表示，（b）具有最小间断序列（1，3，11）的所有八阶级数的共轭表示;以及（c）Klein四群在2处嵌入到Nottingham群中。我们研究的复杂性，新的例子从代数几何性质的方程，他们满足。为此，我们概括的稀疏性理论的幂级数的复杂性，我们给伽罗瓦理论和组合的描述的四个步骤的层次。我们确定我们的不同系列在这个层次结构中的位置。我们构造稀疏的代表共轭类的元素的顺序和深度$2^μ\pm 1$ $(μ\geq 1)$。具有小状态复杂度的系列可以在层次结构中处于高位。例如，对于我们发现的一个新的自动机来说，这是真的，它代表了一个具有5个状态的4阶序列，这是这样一个序列的最小可能数量。
英文摘要：
The Nottingham group at 2 is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \cdots$ in the variable $t$ with coefficients $a_i$ from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest $d\geq 1$ for which $a_2=\dots=a_d=0$. Only a handful of power series of finite order are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series. Up to conjugation, there are only finitely many series $σ$ of order $2^n$ with fixed break sequence (i.e. the sequence of depths of $σ^{\circ 2^i}$). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2. We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth $2^μ\pm 1$ $(μ\geq 1)$. Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states, the minimal possible number for such a series.
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