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 |
| Summary: | 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. |
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| ISSN: | 2271-2097 |