Split absolutely irreducible integer-valued polynomials over discrete valuation domains

TL;DR

This paper characterizes absolutely irreducible integer-valued polynomials over discrete valuation domains, linking them to combinatorial balanced sets and providing criteria for their irreducibility.

Contribution

It offers a complete, constructive characterization of absolutely irreducible split integer-valued polynomials using combinatorial properties of root sets.

Findings

Absolutely irreducible polynomials correspond bijectively to balanced sets.

Existence of a unique multiplicity vector and constant for each balanced set.

Provides criteria for absolute irreducibility over Dedekind domains.

Abstract

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain $(R,M)$ with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements. We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call \emph{balanced}, characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of $M$. For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors 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.