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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| Format: | Article |
| Language: | English |
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Journal of Algebra 633 (2023) 696–72
2023
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| Online Access: | http://hdl.handle.net/20.500.12493/1339 |
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| author | Hiebler, Moritz Nakato, Sarah Roswitha,Rissner |
| author_facet | Hiebler, Moritz Nakato, Sarah Roswitha,Rissner |
| 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 F k , 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 for k, one of which only depends on the valuation of the
denominator of F and the size of the residue class field of R. |
| format | Article |
| id | oai:idr.kab.ac.ug:20.500.12493-1339 |
| institution | KAB-DR |
| language | English |
| publishDate | 2023 |
| publisher | Journal of Algebra 633 (2023) 696–72 |
| record_format | dspace |
| spelling | oai:idr.kab.ac.ug:20.500.12493-13392024-01-17T04:44:54Z Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains Hiebler, Moritz Nakato, Sarah Roswitha,Rissner 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 F k , 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 for k, one of which only depends on the valuation of the denominator of F and the size of the residue class field of R. Kabale University 2023-08-14T07:52:09Z 2023-08-14T07:52:09Z 2023-08-12 Article http://hdl.handle.net/20.500.12493/1339 en application/pdf Journal of Algebra 633 (2023) 696–72 |
| spellingShingle | Hiebler, Moritz Nakato, Sarah Roswitha,Rissner 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 |
| url | http://hdl.handle.net/20.500.12493/1339 |
| work_keys_str_mv | AT hieblermoritz characterizingabsolutelyirreducibleintegervaluedpolynomialsoverdiscretevaluationdomains AT nakatosarah characterizingabsolutelyirreducibleintegervaluedpolynomialsoverdiscretevaluationdomains AT roswitharissner characterizingabsolutelyirreducibleintegervaluedpolynomialsoverdiscretevaluationdomains |