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

TL;DR

This paper estimates the count of monic admissible polynomials with bounded coefficients, providing bounds for both all admissible and irreducible admissible polynomials of degree n.

Contribution

It establishes new asymptotic bounds for the number of monic admissible and irreducible admissible polynomials with coefficients in a fixed range.

Findings

Derived bounds for the total number of admissible polynomials.

Provided lower bounds for the count of irreducible admissible polynomials.

Quantified the growth rate of admissible polynomials as coefficient bounds increase.

Abstract

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}