论文标题
诺丁汉集团的自动机和有限订单元素
Automata and finite order elements in the Nottingham group
论文作者
论文摘要
诺丁汉组为2是（正式）功率系列$ t+ a_2 t^2+ a_3+ a_3 t^3+ \ cdots $在变量$ t $中，带有系数的$ a_i $，带有两个元素，其中组操作由Power of Power of Power of Power of Power Series。此类系列的深度是最大的$ d \ geq 1 $，$ a_2 = \ dots = a_d = 0 $。只有少数功率系列的有限顺序是通过其系数的公式明确知道的。我们在本文中认为，通过自动机以封闭的计算形式描述此类系列是有利的，这是基于Christol定理识别代数和自动系列的有效版本的证明。为了结构轭，只有有限的许多系列$σ$ $ 2^n $具有固定的断裂序列（即$σ^{\ circc 2^i} $的深度序列）。从Witt Vector或Carlitz模块构造开始，我们给出了一个明确的自动机理论描述：（a）代表对所有订单4系列4的共轭，均具有断裂序列（1，m），用于m <10; （b）代表以最小的断裂序列进行所有系列8的连接（1,3,11）； （c）将克莱因四组嵌入到诺丁汉组中。我们从它们所满足的方程式的代数几何特性中研究了新示例的复杂性。为此，我们将功率序列的稀疏理论推广到复杂性的四步层次结构，为此我们既给出了Galois理论和组合描述。我们确定不同系列适合该层次结构的位置。我们为订单二和深度$ 2^μ\ pm 1 $ $（μ\ geq 1）$的元素元素的共轭类别构建稀疏的代表。具有小状态复杂性的系列最终可能会在层次结构中升高。例如，对于我们发现的新自动机来说，这是事实，代表了一系列具有5个状态的顺序4，这是该系列的最小数字。
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.
