## Abstract

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.

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.

Original language | English |
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Number of pages | 26 |

Publication status | Submitted - 29 Jul 2021 |