The irreducibility of weighted sums of polynomials whose irreducible factors are cyclotomic (Q2880118)

For \(n\geq 1\) define \(\gamma_n(x)=\sum_{i=0}^n x^i\). For any \(k\geq 2\), let \(S=\{n_1,n_2,\ldots,n_k\}\) be the set of integers such that \(0<n_1<n_2< \ldots < n_k\), and define the polynomial \(\Gamma_{w(S)}(x):=\sum_{i=1}^k \varepsilon_ix^{w_i}\gamma_{n_i}(x)\), where each \(w_i\) is a non-negative integer and \(\varepsilon_i\in \{-1,1\}\). The main results of the paper concern the irreducibility over the rationals of \(\Gamma_{w(S)}(x)\) in two special cases, namely the case when \(\varepsilon_i=1\) and \(w_i=0\) for all \(i\) and the case when \(k=2\) and \(\varepsilon_1=\varepsilon_2=1\), and exactly one of \(w_1,w_2\) is zero.NEWLINENEWLINEIn the first case the authors show. e.g., that if \(k\) is prime then \(\Gamma_{w(S)}(x)\) is irreducible over the rationals if and only if \(\gcd(n_1+1,n_2+1,\ldots,n_k+1)=1\).NEWLINENEWLINEThe polynomials in the second case form a subclass of the polynomials of the form \(h(x):=x^nf(x)+g(x)\). The irreducibility over the rationals of these polynomials has been studied by, e.g., Filaseta, Ford, Konyagin and Schinzel. In the more restrictive case the present authors consider they get a very simple sufficient criterion for the irreducibility over the rationals.