## 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).

### MSC:

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

Zbl 0171.00701