On an irreducibility theorem of A. Schinzel associated with coverings of the integers. (English) Zbl 0966.11046

A. Schinzel [Acta Arith. 13, 91-101 (1967; Zbl 0171.00701)] found connections between some unproved conjectures on covering systems of congruences and the fact that for every polynomial \(f(x)\) with integer coefficients and \(f(0)\neq 1\), \(f(1)\neq-1\) and \(f(x)\not\equiv 1\), there exists an arithmetic progression \(N\) such that if \(\nu\in N\) then \(x^\nu+f(x)\) is irreducible over the rationals. In the present paper the authors prove extensions of this statement using an alternative approach not depending on covering systems.
Given a polynomial \(F(x)\) its reciprocal is defined by \(x^{\deg(F)}F(1/x)\), and the non–reciprocal of \(F\) is defined as the quotient of \(F\) with the product of all its irreducible reciprocal factors in \({\mathbb Z}[x]\) having positive leading coefficient to the multiplicity they occur as a factor of \(F\).
The first of two main results supposes that
(i) if \(F(x)=\sum_{j=0}^r a_jx^{d_j}\) with \(0=d_0<d_1<\dots<d_r\) and \(a_0a_1\dots a_r\neq 0\) is a polynomial with \(\deg(F)\geq M\), where \(M\) is an explicitly given bound having the form of a doubly exponential sum depending on the integral coefficients of \(F\),
(ii) \(k_0\geq 2\) is a real number,
(iii) the non–reciprocal part of \(F\) is reducible in \({\mathbb Z}[x]\), then there exists a positive integer \(k\in[k_0,\deg(F)]\) such that the polynomial \(G(x,y)=\sum_{j=0}^{\deg(F)|}a_jx^{\delta_j}y^{\lambda_j}\) is reducible in \({\mathbb Z}[x,y]\), where \(\delta_j\equiv d_j \pmod k\) and \(d_j=k\lambda_j+\delta_j\).
In the second main result an extra power of \(x\) factor in the statement of the above result decreases the double exponential bound on the size of \(\deg(f)\) to an exponential one.
The proof argument associates the reducibility of the non–reciprocal part with an elementary problem on distribution of integers in residue classes. (The items 2 and 3 of the references should be attributed to A. Schinzel).


11R09 Polynomials (irreducibility, etc.)
11A07 Congruences; primitive roots; residue systems
11B25 Arithmetic progressions


Zbl 0171.00701