On the nearest irreducible lacunary neighbour to an integer polynomial

Bera, Ranjan and Banerjee, Pradipto (2020) On the nearest irreducible lacunary neighbour to an integer polynomial. Colloquium Mathematicum, 162 (1). pp. 121-134. ISSN 0010-1354

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Abstract

There is an absolute constant D0 > 0 such that if f(x) is an integer polynomial, then there is an integer λ with |λ| ≤ D0 such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. Furthermore, if deg f ≤ 25, then there is a λ with λ ∈ {−2, −1, 0, 1, 2, 3} such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pál Turán.

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