Suppose that $p$ is a prime, and $f(x)$ is a polynomial with integer coefficients and positive degree. Then there exists an integer $n_p$ such that $p | f(n_p)$ if and only if $f(x)$ has a linear factor mod $p$; that is, $f(x) \equiv (x - r_p) g(x) \pmod{p}$ for some polynomial $g(x) \in \mathbb{Z}[x]$ and some integer $r_p$. If this is the case, say $f$ is $p$-soluble.

Let $\mathcal{P}$ be a finite set of primes, say $\mathcal{P} = \{p_1, \cdots, p_t\}$. Say $f$ is $\mathcal{P}$-soluble if $f$ is $p_j$-soluble for $j = 1, \cdots, t$. Define the set

$$\displaystyle A_{\mathcal{P},n}(N) = \{f(x) = a_n x^n + \cdots + a_0 : f \text{ is } \mathcal{P}\text{-soluble}, |a_j| \leq N \forall 0 \leq j \leq n \}.$$

Is there an asymptotic formula known for $\# A_{\mathcal{P},n}$?