2021/07/29 by Sophie Frisch, Frisch, Sophie, Sarah Nakato +3
Mathematics · #11C08 #11R09 #11S05 #13A05 #13B25 #13F20 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2107.14276
openalex publication_date 2021/07/29 · openalex created_date 2022/09/06 · openalex updated_date 2026/07/28
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 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.