The distribution of divisors of polynomials

Let 𝐹(𝑥) be an irreducible polynomial with integer coefficients and degree at least 2. For 𝑥⩾𝑧⩾𝑦⩾2, denote by 𝐻𝐹(𝑥,𝑦,𝑧) the number of integers 𝑛⩽𝑥 such that 𝐹(𝑛) has at least one divisor d with 𝑦<𝑑⩽𝑧. We determine the order of magnitude of 𝐻𝐹(𝑥,𝑦,𝑧) uniformly for 𝑦+𝑦/log𝐶𝑦<𝑧⩽𝑦2 and 𝑦⩽𝑥1−𝛿, showing that the order is the same as the order of 𝐻(𝑥,𝑦,𝑧), the number of positive integers 𝑛⩽𝑥 with a divisor in (𝑦,𝑧]. Here C is an arbitrarily large constant and 𝛿>0 is arbitrarily small.