Abstract
Let ℛ n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ n (t)| of the set ℛ n (t) by showing that, as t → ∞, t 2 log t ≪ |ℛ2(t)| ≪ t 2 log t and t n ≪ |ℛ n (t)| ≪ t n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ k,n (t) ⊂ ℛ n (t) such that p(X) ∈ ℛ k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t k+1 ≪ |ℛ k,n (t)| ≪ t k+1 and hence |ℛ n−1,n (t)| ≫ |ℛ n (t)| so that ℛ n−1,n (t) is the dominating subclass of ℛ n (t) since we can show that |ℛ n (t)∖ℛ n−1,n (t)| ≪ t n−1(log t)2.On the contrary, if R s n (t) is the total number of all polynomials in ℛ n (t) which split completely into linear factors over ℤ, then t 2(log t)n−1 ≪ R s n (t) ≪ t 2 (log t)n−1 (t → ∞) for every fixed n ≥ 2.
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(Communicated by Stanislav Jakubec)
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Kuba, G. On the distribution of reducible polynomials. Math. Slovaca 59, 349–356 (2009). https://doi.org/10.2478/s12175-009-0131-6
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DOI: https://doi.org/10.2478/s12175-009-0131-6