Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains.

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 F k , 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 for k , one of which only depends on the valuation of the denominator of F and the size of the residue class field of R. [ABSTRACT FROM AUTHOR]