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

2019/12/22 by Sophie Frisch, Frisch, Sophie, Sarah Nakato +1
Computer Science · Mathematics · Medicine · #11C08 #11R09 #13A05 #13B25 #13F20 #13P05 #Commutative Algebra (math.AC) #FOS: Mathematics #Magnolia and Illicium research #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1912.10535

openalex publication_date 2019/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Int(D)=\f∈ K[x]| f(D)⊆ 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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