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


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Apostol T.M.: Introduction to Analytic Number Theory. Springer, New York (1998)zbMATHGoogle Scholar
  2. 2.
    Chela R.: Reducible polynomials. J. London Math. Soc. 38, 183–188 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Chern S.-J., Vaaler J.D.: The distribution of values of Mahler’s measure. J. Reine Angew. Math. 540, 1–47 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dörge K.: Abschätzung der Anzahl der reduziblen Polynome. Math. Ann. 160, 59–63 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Dubickas A.: Polynomials irreducible by Eisenstein’s criterion. Appl. Algebra Eng. Commun. Comput. 14, 127–132 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Fel’dman, N.I.: Approximations of algebraic numbers. Moskov. Gos. Univ., Moscow, (in Russian) (1981)Google Scholar
  7. 7.
    Heyman R., Shparlinski I.E.: On the number of Eisenstein polynomials of bounded height. Appl. Algebra Eng. Commun. Comput. 24, 149–156 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Knobloch H.W.: Zum Hilbertschen Irreduzibilitätssatz. Abh. Math. Sem. Univ. Hamburg 19, 176–190 (1955)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Konyagin S.V.: On the number of irreducible polynomials with 0,1 coefficients. Acta Arith. 88, 333–350 (1999)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Koyuncu F., Özbudak F.: Probabilities for absolute irreducibility of multivariate polynomials by the polytope method. Turkish J. Math. 35, 367–377 (2011)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Kuba G.: On the distribution of reducible polynomials. Math. Slovaca 59, 349–356 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Masser, D., Vaaler, J.D.: Counting algebraic numbers with large height. I. In: Schlickewei, Hans Peter et al. (eds.) Diophantine approximation. Festschrift for Wolfgang Schmidt, Vienna, Austria, 2003, Springer, Developments in Mathematics, 16, 237–243 (2008)Google Scholar
  13. 13.
    Masser D., Vaaler J.D.: Counting algebraic numbers with large height. II. Trans. Am. Math. Soc. 359, 427–455 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Peter M.: Lattice points in convex bodies with planar points on the boundary. Monatsh. Math. 135, 37–57 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Pólya G., Szegö G.: Problems and Theorems in Analysis, Vol II. Springer, Berlin, Heidelberg, New York (1976)CrossRefGoogle Scholar
  16. 16.
    Schanuel S.H.: Heights in number fields. Bull. Soc. Math. France 107, 433–449 (1979)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Schinzel A.: Polynomials with special regard to irreducibility. CUP, Cambridge (2000)CrossRefGoogle Scholar
  18. 18.
    Schmidt W.M.: Northcott’s theorem on heights I. A general estimate. Monatsch. Math. 115, 169–181 (1993)CrossRefzbMATHGoogle Scholar
  19. 19.
    van der Waerden B.L.: Die Seltenhen der Gleichungen mit Affekt. Math. Ann. 109, 13–16 (1934)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Widmer M.: Counting primitive points of bounded height. Trans. Am. Math. Soc. 362, 4793–4829 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

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

Personalised recommendations