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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 31, 2015 12:00–12:25, Moscow, Steklov Mathematical Institute of the Russian Academy of Sciences  Exact bounds for the number of integer polynomials with bounded discriminants (common paper with N. Budarina and F. Goetze)

V. I. Bernik

Institute of Mathematics of the National Academy of Sciences of Belarus

Abstract: For a polynomial $P(x) = a_n x^n + …+ a_1 x + a_0$ of an arbitrary degree $n$ and height $H = H(P) = \max_{0\le i \le n}|a_i|$ with complex roots $\alpha_1 , …, \alpha_n$, 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) = \{ P \in \mathbb{Z}[x] \: : \: \deg P = n, \quad H(P) \le Q, \quad 0 < |D| < Q^{2n-2-2v} \} . \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.
 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.
 V. Beresnevich, “Rational points near manifolds and metric Diophantine approximation”, Ann. Math. (2), 175:1 (2012), 187–-235.

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