Polynomial Sequences in Groups
Polynomial Sequences in Groups
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DOI：
10.1006/jabr.1997.7269
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发表时间：
1998-03
影响因子：
0.9
通讯作者：
A. Leibman
中科院分区：
文献类型：
--
作者：
A. Leibman
摘要：给定一个低中心序列g = G1的群g，我们说任意位置的序列g: Z→吉斯多项式，使得dg (n) = g(n)−1g（n + 1）次的差算子得到的序列取其值inGk。引入多项式序列的阶的概念，证明了不超过给定阶的多项式序列构成一个群。作为应用，我们得到了Hall-Petresco定理的以下推广：定理。LetG = G1·G3为群g的下中心级数，设x∈Gk，y∈Gland letp，分别为k度和l度的多项式Z→Z。那么存在一个序列z0∈G，zi∈Gifori∈N，使得对于所有N∈N。
Abstract Given a groupGwith lower central seriesG = G1 ⊇ G2 ⊇ G3 ⊇ ···, we say that a sequenceg: Z → Gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatorDg(n) = g(n) − 1g(n + 1)dtimes takes its values inGk. We introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group. As an application we obtain the following extension of the Hall–Petresco theorem: THEOREM.LetG = G1 ⊇ G2 ⊇ G3 ⊇ ···be the lower central series of a group G.Let x ∈ Gk,y ∈ Gland letp, qbe polynomials Z → Z of degrees k and l,respectively. Then there is a sequencez0 ∈ G,zi ∈ Gifori ∈ N ,such thatfor all n ∈ N .