Exact bounds for the number of integer polynomials with bounded discriminants (common paper with N. Budarina and F. Goetze)

For a polynomial $ P(x) = a_n x^n + \dots + a_1 x + a_0$ of an arbitrary degree $n$ and height $H = H(P) = \max_{0\le i \le n}|a_i|$ with complex roots $\alpha_1 , \dots, \alpha_n$, the discriminant $D(P)$ is defined as follows:
\begin{equation} D = D(P) = a_n^{2n-2} \prod_{1 \le i < j \le n} (\alpha_i - \alpha_j)^2. \end{equation}
It is well known that the discriminant $D$ can also be calculated as a determinant of a certain $(2n-1)\times(2n-1)$ integer matrix.
For a positive integer $Q>1$ and a real constant $v$, $0 \le v \le n-1$, let us define the following class of integer polynomials:
\begin{equation} \mathcal{P}_n(Q,v) = \left\{ P \in \mathbb{Z}[x] \: : \: \deg P = n, \quad H(P) \le Q, \quad 0 < |D| < Q^{2n-2-2v} \right\} . \end{equation}
Estimating the number of polynomials $\#\mathcal{P}_n(Q,v)$ in this class is highly relevant to many problems in Diophantine approximation [1, 2].
During the talk, we are going to present a number of upper and lower bounds for $\# \mathcal{P}_n(Q,v)$ based on metric theory of Diophantine approximation of dependent variables. Both well-established and recent results will be discussed.
[1] V.I. Bernik, “A metric theorem on the simultaneous approximation of a zero by the values of integral polynomials”, Izv. Akad. Nauk SSSR, Ser. Mat., 44:1 (1980), 24–-45.
[2] V. Beresnevich, “Rational points near manifolds and metric Diophantine approximation”, Ann. Math. (2), 175:1 (2012), 187–-235.