New estimates for the number of integer polynomials with given discriminants (Q779832)

The present article deals with an upper bound for the number of polynomials \(P\in \mathcal P^{'} _4 (Q,v)\). Here \(\mathcal P^{'} _4 (Q,v)\) is the set of polynomials \(P\in \mathcal P_4 (Q,v)\) with distinct roots satisfying certain conditions. The authors present a new method of upper bounds for the number of integer polynomials of the fourth degree with a given discriminant. Connections between the present research and known results are described. A brief survey is devoted to the problem of counting polynomials with a small discriminant \(D(P)\) in the theory of Diophantine approximations. All auxiliary notions are recalled, and some statements which are useful for proving the main result are given.