Split absolutely irreducible integer-valued polynomials over discrete valuation domains

We investigate non-unique factorization of integer-valued poly-
nomials over discrete valuation domains with finite residue field. There exist
non-absolutely irreducible elements, that is, irreducible elements whose powers
have other factorizations into irreducibles than the obvious one. We completely
and constructively characterize the absolutely irreducible elements among split
integer-valued polynomials. They correspond bijectively to finite sets with a
certain property regarding M -adic topology. For each such “balanced” set of
roots, there exists a unique vector of multiplicities and a unique constant so
that the corresponding product of monic linear factors with multiplicities times
the constant is an absolutely irreducible integer-valued polynomial. This also
yields sufficient criteria for integer-valued polynomials over Dedekind domains
to be absolutely irreducible.