Irreducible polynomials in Int(ℤ)
In order to fully understand the factorization behavior of the ring Int(ℤ) = {f ∈ ℚ[x] | f (ℤ) ⊆ ℤ} of integer-valued polynomials on ℤ, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to recognize whether an integer-valued polynomial g/d] $P_{\tex...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
EDP Sciences
2018-01-01
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| Series: | ITM Web of Conferences |
| Online Access: | https://doi.org/10.1051/itmconf/20182001004 |
| _version_ | 1818912505252020224 |
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| author | Antoniou Austin Nakato Sarah Rissner Roswitha |
| author_facet | Antoniou Austin Nakato Sarah Rissner Roswitha |
| author_sort | Antoniou Austin |
| collection | DOAJ |
| description | In order to fully understand the factorization behavior of the ring Int(ℤ) = {f ∈ ℚ[x] | f (ℤ) ⊆ ℤ} of integer-valued polynomials on ℤ, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to recognize whether an integer-valued polynomial
g/d]
$P_{\textrm{rad}} \propto P_{\textrm{sw}}^{1.2}$
gd
is irreducible in the case where d is a square-free integer and g ∈ ℤ[x] has fixed divisor d. For integer-valued polynomials with arbitrary composite denominators, so far there is no algorithmic criterion known to recognize whether they are irreducible. We describe a computational method which allows us to recognize all irreduciblexc polynomials in Int(ℤ). We present some known facts, preliminary new results and open questions. |
| first_indexed | 2024-12-19T23:15:40Z |
| format | Article |
| id | doaj.art-354ee20e82c54ad195059bf3178ebe15 |
| institution | Directory Open Access Journal |
| issn | 2271-2097 |
| language | English |
| last_indexed | 2024-12-19T23:15:40Z |
| publishDate | 2018-01-01 |
| publisher | EDP Sciences |
| record_format | Article |
| series | ITM Web of Conferences |
| spelling | doaj.art-354ee20e82c54ad195059bf3178ebe152022-12-21T20:02:06ZengEDP SciencesITM Web of Conferences2271-20972018-01-01200100410.1051/itmconf/20182001004itmconf_icm2018_01004Irreducible polynomials in Int(ℤ)Antoniou AustinNakato SarahRissner RoswithaIn order to fully understand the factorization behavior of the ring Int(ℤ) = {f ∈ ℚ[x] | f (ℤ) ⊆ ℤ} of integer-valued polynomials on ℤ, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to recognize whether an integer-valued polynomial g/d] $P_{\textrm{rad}} \propto P_{\textrm{sw}}^{1.2}$ gd is irreducible in the case where d is a square-free integer and g ∈ ℤ[x] has fixed divisor d. For integer-valued polynomials with arbitrary composite denominators, so far there is no algorithmic criterion known to recognize whether they are irreducible. We describe a computational method which allows us to recognize all irreduciblexc polynomials in Int(ℤ). We present some known facts, preliminary new results and open questions.https://doi.org/10.1051/itmconf/20182001004 |
| spellingShingle | Antoniou Austin Nakato Sarah Rissner Roswitha Irreducible polynomials in Int(ℤ) ITM Web of Conferences |
| title | Irreducible polynomials in Int(ℤ) |
| title_full | Irreducible polynomials in Int(ℤ) |
| title_fullStr | Irreducible polynomials in Int(ℤ) |
| title_full_unstemmed | Irreducible polynomials in Int(ℤ) |
| title_short | Irreducible polynomials in Int(ℤ) |
| title_sort | irreducible polynomials in int z |
| url | https://doi.org/10.1051/itmconf/20182001004 |
| work_keys_str_mv | AT antoniouaustin irreduciblepolynomialsinintz AT nakatosarah irreduciblepolynomialsinintz AT rissnerroswitha irreduciblepolynomialsinintz |