On the number of monic admissible polynomials in the ring $\mathbb{Z}[x]$

In this paper we study admissible polynomials. We establish an estimate for the number of admissible polynomials of degree $n$ with coeffients $a_i$ satisfying $0\leq a_i\leq H$ for a fixed $H$, for $i=0,1,2, \ldots, n-1$. In particular, letting $\mathcal{N}(H)$ denotes the number of monic admissible polynomials of degree $n\geq 3$ with coefficients satisfying the inequality $0\leq a_i\leq H$, we show that \begin{align}\frac{H^{n-1}}{(n-1)!}+O(H^{n-2})\leq \mathcal{N}(H) \leq \frac{n^{n-1}H^{n-1}}{(n-1)!}+O(H^{n-2}).\nonumber \end{align} Also letting $\mathcal{A}(H)$ denotes the number of monic irreducible admissible polynomials, with coefficients satisfying the same condition , we show that \begin{align}\mathcal{A}(H)\geq \frac{H^{n-1}}{(n-1)!}+O\bigg( H^{n-4/3}(\log H)^{2/3}\bigg).\nonumber \end{align}