Polynomial sequences in groups
Let \(\mathbb{Z}_+=\{0,1,2,\dots\}\), \(\mathbb{Z}_*=\{-\infty,0,1,2,\dots\}\) and \((-\infty)+(-\infty)=-\infty\), \(-\infty<t\), \((-\infty)\pm t=(-\infty)\) for every \(t\in\mathbb{Z}_+\). Let \(G\) be a group and \(G=G_1\geq G_2\geq\cdots\geq G_n\geq\cdots\) the lower central series of \(G\). If \(g\colon\mathbb{Z}\to G\) is a sequence then its derivative \(Dg\) is the sequence defined by \(Dg(n)=g(n)^{-1}g(n+1)\). The derivation \(D\) is a mapping from the set \(G^\mathbb{Z}\) into itself: let \(D^1=D\), \(D^{t+1}=D\circ D^t\), \(t=1,2,\dots\), and \(D^{-\infty}=D^0=id_G\mathbb{Z}\). Let \(\overline d=(d_1,d_2,\dots)\) where \(d_k\in\mathbb{Z}_*\). A sequence \(g\in G^\mathbb{Z}\) is said to be polynomial of degree \(\leq\overline d\) if for every \(k\in\mathbb{N}\), \(D^{d_k+1}g\) takes its values in \(G_{k+1}\): \(D^{d_k+1}g(n)\in G_{k+1}\) for all \(n\in\mathbb{Z}\). A sequence \(\overline d=(d_k)_{k\in\mathbb{N}}\) with \(d_k\in\mathbb{Z}_*\) is said to be superadditive if it is nondecreasing and satisfies \(d_i+d_j\leq d_{i+j}\) for all \(i,j\in\mathbb{N}\). The following theorem is the main result of this paper. Theorem 1.12. Let \(\overline d\) be a superadditive sequence. The polynomial sequences of degree \(\leq\overline d\) form a group (with respect to elementwise multiplication). Corollary 1.13. The set of all polynomial sequences in \(G\) is a group. Corollary 3.3. Let \(x_1,\dots,x_s\) be elements of \(G\) where \(x_j\in G_{k_j}\), and let \(p_1,\dots,p_s\) be integral polynomials with \(\deg p_j\leq k_j\), \(1\leq j\leq s\). Let \(S\) be the set of nonnegative integers with a fixed linear ordering. Then there are \(z_0\in G\) and \(z_k\in G_k\) for \(k=1,2,\dots\) such that \[ \prod_{1\leq j\leq s} x^{p_j(n)}_j=\prod_{\substack{ k\in S\\ 0\leq k\leq n}} z^{{n\choose k}}_k \] for all \(n\in\mathbb{Z}_+\). If the ordering of \(S\) is standard, the last product is \(\prod_{0\leq k\leq n} z^{{n\choose k}}_k\). If, in addition, \(p_j(0)=\cdots=p_j(\ell)=0\) for all \(j=1,\dots,s\), then \[ \prod_{1\leq j\leq s} x^{p_{ij}(n)}_{ij}=\prod_{\substack{ k\in S\\ \ell+1\leq k\leq n}} z^{{n\choose k}}_k \] for all \(n\in\mathbb{Z}_+\).
scientific article; zbMATH DE number 1211660