A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator

TL;DR

This paper introduces a graph-theoretic criterion to determine absolute irreducibility of integer-valued polynomials over principal ideal domains, providing a practical method with conditions that are both sufficient and necessary in certain cases.

Contribution

It presents a novel graph-theoretic criterion for absolute irreducibility of integer-valued polynomials, especially effective for those with square-free denominators.

Findings

The criterion is easy to verify in practice.

It is sufficient for absolute irreducibility.

It is necessary and sufficient for polynomials with square-free denominators.

Abstract

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$, of integer-valued polynomials on a principal ideal domain $D$ with quotient field $K$, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator.