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Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 439–456 | Cite as

On the number of reducible polynomials of bounded naive height

  • Artūras DubickasEmail author
Article

Abstract

We prove an asymptotical formula for the number of reducible integer polynomials of degree d and of naive height at most T when \({T \to \infty}\). The main term turns out to be of the form \({\kappa_d T^d}\) for each \({d \geq 3}\), where the constant \({\kappa_d}\) is given in terms of some infinite Dirichlet series involving the volumes of symmetric convex bodies in \({\mathbb{R}^d}\). For d = 2, we prove that there are asymptotically \({\kappa_2 T^2 \,\text{log} T}\) of such polynomials, where \({\kappa_2:=6(3\sqrt{5}+2\,\text{log} (1+\sqrt{5}) -2 \,\text{log}\, 2)/\pi^2}\). Earlier results in this direction were given by van der Waerden, Pólya and Szegö, Dörge, Chela, and Kuba.

Mathematics Subject Classification (2000)

11R09 12E05 

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsVilnius UniversityVilniusLithuania
  2. 2.Institute of Mathematics and InformaticsVilnius UniversityVilniusLithuania

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