Further irreducibility criteria for polynomials with non-negative coefficients (Q2833594)

The article deals with implications between a polynomial \(f(x)\in\mathbb Z[x]\) being irreducible and \(f(b)\) being prime for some \(b\in\mathbb Z\).NEWLINENEWLINEMore precisely, suppose \(d_n d_{n-1}\dots d_0\) is the representation of a positive integer in base 10. It was shown by \textit{G. Pólya} and \textit{G. Szegö} [Aufgaben und Lehrsätze aus der Analysis. 3. bericht. Aufl. Berlin: Springer (1964; Zbl 0122.29704)] that if that integer is prime, then \(d_n x^n + \ldots + d_1 x + d_0 \in {\mathbb Z}[x]\) is irreducible over \(\mathbb Q\).NEWLINENEWLINE\textit{M. Filaseta} and \textit{S. Gross} [J. Number Theory 137, 16--49 (2014; Zbl 1285.11053)] generalized this as follows. Suppose \(f(x)\) has nonnegative coefficients bounded by an explicitly given constant and \(f(10)\) is prime, then \(f(x)\) is irreducible over \({\mathbb Q}\). They also showed that the bound is sharp.NEWLINENEWLINEThe main goal of the authors is the extension of the results in [Filaseta and Gross, loc. cit.] to different bases. It turns out that the smaller bases are more difficult to handle.NEWLINENEWLINEFor a statement of the main result, let \(\Phi_n(x)\) denote the \(n\)th cyclotomic polynomial.NEWLINENEWLINETheorem 1.1. Fix an integer \(b\) in the interval \([2,20]\), let \(f(x)\in\mathbb Z[x]\) have nonnegative coefficients and assume \(f(b)\) is prime. If the coefficients are smaller than a bound depending on \(b\), then \(f(x)\) is irreducible over \(\mathbb Q\). Similarly, for other bounds on the coefficients and smaller ranges of \(b\), the polynomial \(f(x)\) is irreducible or divisible by a cyclotomic polynomial \(\Phi_3(x-b)\) or \(\Phi_4(x-b)\). For \(4\leq b \leq 20\), the bounds are sharp.NEWLINENEWLINEFor the proofs, the authors use the same three main methods as in [Filaseta and Gross, loc. cit.].NEWLINENEWLINETheorem 2.2 contains a smaller result with a short proof which demonstrates some of the ideas.