Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
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Date
2024
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Abstract
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
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Keywords
Non-unique factorizatio, Irreducible elements Absolutely irreducible elements Integer-valued polynomials