Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
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...
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2024
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| Online Access: | http://hdl.handle.net/20.500.12493/1947 |
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| author | Hiebler, Moritz Nakato, Sarah Rissner, Roswitha |
| author_facet | Hiebler, Moritz Nakato, Sarah Rissner, Roswitha |
| author_sort | Hiebler, Moritz |
| collection | KAB-DR |
| description | 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 |
| id | oai:idr.kab.ac.ug:20.500.12493-1947 |
| institution | KAB-DR |
| publishDate | 2024 |
| record_format | dspace |
| spelling | oai:idr.kab.ac.ug:20.500.12493-19472024-02-09T00:00:29Z Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains Hiebler, Moritz Nakato, Sarah Rissner, Roswitha Non-unique factorizatio Irreducible elements Absolutely irreducible elements Integer-valued polynomials 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 Kabale University 2024-02-08T08:53:30Z 2024-02-08T08:53:30Z 2024 http://hdl.handle.net/20.500.12493/1947 Attribution-NonCommercial-NoDerivs 3.0 United States http://creativecommons.org/licenses/by-nc-nd/3.0/us/ application/pdf |
| spellingShingle | Non-unique factorizatio Irreducible elements Absolutely irreducible elements Integer-valued polynomials Hiebler, Moritz Nakato, Sarah Rissner, Roswitha Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |
| title | Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |
| title_full | Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |
| title_fullStr | Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |
| title_full_unstemmed | Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |
| title_short | Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains |
| title_sort | characterizing absolutely irreducible integer valued polynomials over discrete valuation domains |
| topic | Non-unique factorizatio Irreducible elements Absolutely irreducible elements Integer-valued polynomials |
| url | http://hdl.handle.net/20.500.12493/1947 |
| work_keys_str_mv | AT hieblermoritz characterizingabsolutelyirreducibleintegervaluedpolynomialsoverdiscretevaluationdomains AT nakatosarah characterizingabsolutelyirreducibleintegervaluedpolynomialsoverdiscretevaluationdomains AT rissnerroswitha characterizingabsolutelyirreducibleintegervaluedpolynomialsoverdiscretevaluationdomains |