Advertisement

Periodica Mathematica Hungarica

, Volume 78, Issue 1, pp 98–109 | Cite as

Bounding univariate and multivariate reducible polynomials with restricted height

  • Artūras DubickasEmail author
Article

Abstract

Let \(d,H \geqslant 2\), \(m, u \geqslant 0\) be some integers satisfying \(m+u \leqslant d\). Consider a set of univariate integer polynomials of degree d whose m coefficients for the highest powers of x and u coefficients for the lowest powers of x are fixed, whereas the remaining \(g=d-m-u+1\) coefficients are all bounded by H in absolute value. We show that among those \((2H+1)^g\) polynomials at most \(c d(2H+1)^{g-1}(\log (2H))^{\delta }\) are reducible over \(\mathbb Q\), where the constant \(c>0\) depends only on two extreme coefficients (if they are fixed) and does not depend on d and H. Here, \(\delta =2\) if \(m=u=0\); \(\delta =1\) if only one of mu is zero; \(\delta =0\) if none of mu is zero. This estimate is better than the previous one in certain range of d and H. We also prove an estimate for the number of integer reducible polynomials in \(n \geqslant 2\) variables of degree \(d \geqslant 1\) in each variable and height at most \(H \geqslant 1\). It is completely explicit in terms of ndH and implies that the probability for such a polynomial to be reducible tends to zero as \(\max (n,d,H) \rightarrow \infty \). The condition \(n \geqslant 2\) is essential in the proof: despite some recent progress the problem in general remains open for \(n=1\).

Keywords

Reducible polynomials Multivariate reducible polynomials Height 

Mathematics Subject Classification

11R09 12D05 12E05 

Notes

Acknowledgements

This research was funded by a Grant (No. S-MIP-17-66/LSS-110000-1274) from the Research Council of Lithuania.

References

  1. 1.
    S. Akiyama, A. Pethő, On the distribution of polynomials with bounded roots I. Polynomials with real coefficients. J. Math. Soc. Jpn. 66, 927–949 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Akiyama, A. Pethő, On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients. Unif. Distrib. Theory 9, 5–19 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    L. Bary-Soroker, G. Kozma, Is a bivariate polynomial with \(\pm 1\) coefficients irreducible? very likely!. Int. J. Number Theory 13, 933–936 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L. Bary-Soroker, G. Kozma, Irreducible polynomials of bounded height, preprint arXiv:1710.05165 (2017)
  5. 5.
    M. Bhargava, J.E. Cremona, T. Fisher, N.G. Jones, J.P. Keating, What is the probability that a random integral quadratic form in n variables has an integral zero? Int. Math. Res. Not. IMRN 12, 3828–3848 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. Calegari, Z. Huang, Counting Perron numbers by absolute value. J. Lond. Math. Soc. 96, 181–200 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Castillo, R. Dietmann, On Hilbert’s irreducibility theorem. Acta Arith. 180, 1–14 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    R. Chela, Reducible polynomials. J. Lond. Math. Soc. 38, 183–188 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S.-J. Chern, J.D. Vaaler, The distribution of values of Mahler’s measure. J. Reine Angew. Math. 540, 1–47 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    K. Dörge, Abschätzung der Anzahl der reduziblen Polynome. Math. Ann. 160, 59–63 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Dubickas, Polynomials irreducible by Eisenstein’s criterion. Appl. Algebra Eng. Commun. Comput. 14, 127–132 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Dubickas, On the number of reducible polynomials of bounded naive height. Manuscr. Math. 144, 439–456 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Dubickas, Counting integer reducible polynomials with bounded measure. Appl. Anal. Discrete Math. 10, 308–324 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    A. Dubickas, M. Sha, Counting and testing dominant polynomials. Exp. Math. 24, 312–325 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    R. Grizzard, J. Gunther, Slicing the stars: counting algebraic integers, and units by degree and height. Algebra Number Theory 11, 1385–1436 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    R. Heyman, I.E. Shparlinski, On the number of Eisenstein polynomials of bounded height. Appl. Algebra Eng. Commun. Comput. 24, 149–156 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    R. Heyman, I.E. Shparlinski, On shifted Eisenstein polynomials. Period. Math. Hung. 69, 170–181 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    S.V. Konyagin, On the number of irreducible polynomials with \(0,1\) coefficients. Acta Arith. 88, 333–350 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    F. Koyuncu, F. Özbudak, Probabilities for absolute irreducibility of multivariate polynomials by the polytope method. Turkish J. Math. 35, 367–377 (2011)MathSciNetzbMATHGoogle Scholar
  20. 20.
    G. Kuba, On the distribution of reducible polynomials. Math. Slovaca 59, 349–356 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    G. Micheli, R. Schnyder, The density of shifted and affine Eisenstein polynomials. Proc. Am. Math. Soc. 144, 4651–4661 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A.M. Odlyzko, B. Poonen, Zeros of polynomials with \(0,1\) coefficients. Enseign. Math. 39, 317–348 (1993)MathSciNetzbMATHGoogle Scholar
  23. 23.
    I. Rivin, Galois groups of generic polynomials, preprint arXiv:1511.06446 (2015)
  24. 24.
    G. Pólya, G. Szegö, Problems and Theorems in Analysis, vol. II (Springer, Berlin, 1976)CrossRefzbMATHGoogle Scholar
  25. 25.
    B.L. van der Waerden, Die Seltenhen der Gleichungen mit Affekt. Math. Ann. 109, 13–16 (1934)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Institute of Mathematics, Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania

Personalised recommendations