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arXiv · 1912.10535

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

Sophie Frisch·Sarah Nakato
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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.

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Sophie Frisch, Sarah Nakato. 2019-12-22. A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator. https://arxiv.org/abs/1912.10535

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