The cyclotomic polynomial topologically (Q2450166)

For any positive integer \(n\) and any primitive \(n^{\mathrm{th}}\) root of unity \(\zeta\), we denote \(c_0\), \(c_1\), \(\dots\), \(c_{\varphi(n)}\) (where \(\varphi\) is the Euler phi function) the coefficients of the \(n^{\mathrm{th}}\) cyclotomic polynomial \[ \Phi_n(x)=\prod_{i \in \left(\mathbb{Z}/n\mathbb{Z}\right)^\times} \left(x-\zeta^i\right) \] The coefficients \(c_i\) are integers and, in this paper, the authors give two interpretations of these coefficients in terms of orders of cyclic groups. The first interpretation is an algebraic one, \(c_j\) being the order of the cyclic group \({\mathbb{Z}}[\zeta]/\mathbb{Z}[A]\) where \(\mathbb{Z}[A]\) is the set of all \(\mathbb{Z}\)-linear combinations of elements of \(A:=\{1,\zeta,\zeta^2,\dots,\zeta^{\varphi(n)}\} \setminus \{\zeta^j\}\). The main results of the paper correspond to the second interpretation which is of topological nature. Let us suppose that the integer \(n\) admits a square free prime factorization \(p_1p_2\dots p_d\). Denoting \(K_i\) a set (or 0-dimensional complex) of \(i\) vertices, the authors consider the simplicial join \(K_{p_1,p_2,\dots,p_d}:= K_{p_1} * K_{p_2} * \dots * K_{p_d}\) (which is a complete \(d\)-partite complex). The maximal faces (or facets) of \(K_{p_1,p_2,\dots,p_d}\) are labelled by residues modulo \(n\) and the Theorem 1 gives the following calculus of reduced integral simplicial homology groups : \[ \widetilde{H}_i(K_{\{j\}},\mathbb Z) = \begin{cases} \mathbb{Z}/c_j\mathbb{Z} &{\text{if }}i=d-2 \\ \mathbb{Z} &{\text{if both }} i=d-1 {\text{ and }}c_j=0 \\ 0&{\text{otherwise}} \end{cases} \] where \(K_{\{j\}}\) is the subcomplex of \(K_{p_1,p_2,\dots,p_d}\) generated by facets labelled by elements of the set of residues \[ \{j, \varphi(n)+1,\varphi(n)+2,\dots, n-1\} \] A second theorem, based on a similar construction \(K_{\{j,j'\}}\) gives information about the signs of the nonzero coefficients of the cyclotomic polynomial. The result stated in Theorem 1 has been extended by \textit{R. Meshulam} [J. Algebr. Comb. 35, No. 4, 565--571 (2012; Zbl 1272.18011)] by using another approach.