Abstract

Let ℛn(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 |ℛn(t)| of the set ℛn(t) by showing that, as t → ∞, t 2 log t ≪ |ℛ2(t)| ≪ t 2 log t and t n ≪ |ℛn(t)| ≪ t n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛk,n(t) ⊂ ℛn(t) such that p(X) ∈ ℛk,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 ≪ |ℛk,n(t)| ≪ t k+1 and hence |ℛn−1,n (t)| ≫ |ℛn(t)| so that ℛn−1,n (t) is the dominating subclass of ℛn(t) since we can show that |ℛn(t)∖ℛn−1,n (t)| ≪ t n−1(log t)2.On the contrary, if R ns(t) is the total number of all polynomials in ℛn(t) which split completely into linear factors over ℤ, then t 2(log t)n−1 ≪ R ns(t) ≪ t 2 (log t)n−1 (t → ∞) for every fixed n ≥ 2.