Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
| dc.contributor.author | Hiebler, Moritz | |
| dc.contributor.author | Nakato, Sarah | |
| dc.contributor.author | Rissner, Roswitha | |
| dc.date.accessioned | 2024-02-08T08:53:30Z | |
| dc.date.available | 2024-02-08T08:53:30Z | |
| dc.date.issued | 2024 | |
| dc.description.abstract | Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one. In this paper, we study irreducible polynomials F ∈ Int(R) where R is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number S ∈ N that reduces the absolute irreducibility of F to the unique factorization of F S. To this end, we establish a connection between the factors of powers of F and the kernel of a certain linear map that we associate to F. This connection yields a characterization of absolute irreducibility in terms of this so-called fixed divisor kernel. Given a non-trivial element v of this kernel, we explicitly construct non-trivial factorizations of Fk, provided that k ≥ L, where L depends on F as well as the choice of v. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds | |
| dc.description.sponsorship | Kabale University | |
| dc.identifier.uri | http://hdl.handle.net/20.500.12493/1947 | |
| dc.rights | Attribution-NonCommercial-NoDerivs 3.0 United States | en |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/3.0/us/ | |
| dc.subject | Non-unique factorizatio | |
| dc.subject | Irreducible elements Absolutely irreducible elements Integer-valued polynomials | |
| dc.title | Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |