Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains

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Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one. In this paper, we study irreducible polynomials F ∈ Int(R) where R is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number S ∈ N that reduces the absolute irreducibility of F to the unique factorization of F S. To this end, we establish a connection between the factors of powers of F and the kernel of a certain linear map that we associate to F. This connection yields a characterization of absolute irreducibility in terms of this so-called fixed divisor kernel. Given a non-trivial element v of this kernel, we explicitly construct non-trivial factorizations of Fk, provided that k ≥ L, where L depends on F as well as the choice of v. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds



Non-unique factorizatio, Irreducible elements Absolutely irreducible elements Integer-valued polynomials