Irreducible polynomials in Int(ℤ) Article Swipe

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Austin Antoniou , Sarah Nakato , Roswitha Rissner ·

YOU? · · 2018 · Open Access · · DOI: https://doi.org/10.1051/itmconf/20182001004 · OA: W2896364880

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 ] 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.