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Journal of Mathematical Sciences

, Volume 219, Issue 5, pp 700–706 | Cite as

Discriminant and Root Separation of Integral Polynomials

  • F. Götze
  • D. Zaporozhets
Article

Consider a random polynomial G Q (x) = ξ Q,n x n  + ξ Q,n − 1 x n − 1 + ⋯ + ξ Q,0 with independent coefficients that are uniformly distributed on 2Q+1 integer points {−Q, . . .,Q}. Denote by D(GQ) the discriminant of GQ. We show that there exists a constant Cn depending on n only such that for all Q ≥ 2, the distribution of D(GQ) can be approximated as follows: \( \underset{-\infty \le a\le b\le -\infty }{ \sup}\left|\mathrm{P}\left(a\frac{D\left({G}_Q\right)}{Q^{2n-2}}\le b\right)-{\displaystyle \underset{a}{\overset{b}{\int }}{\upvarphi}_n(x)dx}\right|\le \frac{C_n}{ \log Q}, \) where \( \varphi \) n denotes the probability density function of the discriminant of a random polynomial of degree n with independent coefficients that are uniformly distributed on [−1, 1]. Let Δ(GQ) denote the minimal distance between complex roots of GQ. As an application, we show that for any ε > 0 there exists a constant δn > 0 such that Δ(GQ) is stochastically bounded from below/above for all sufficiently large Q in the following sense: \( \mathrm{P}\left({\delta}_n<\varDelta \left({G}_Q\right)<\frac{1}{\delta_n}\right)>1-\varepsilon \). Bibliography: 14 titles.

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References

  1. 1.
    V. Beresnevich, V. Bernik, and F. Götze, “The distribution of close conjugate algebraic numbers,” Compos. Math., 146, 1165–1179 (2010).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    V. Beresnevich, V. Bernik, and F. Götze, “Integral polynomials with small discriminants and resultants,” Preprint, arXiv:1501.05767 (2015).Google Scholar
  3. 3.
    V. Beresnevich, V. Bernik, F. Götze, and O. Kukso, “Distribution of algebraic numbers and metric theory of Diophantine approximation,” In: Limit Theorems in Probability, Statistics, and Number Theory, Springer (2013), pp. 23–48.Google Scholar
  4. 4.
    V. Bernik, F. Götze, and O. Kukso, “Lower bounds for the number of integral polynomials with given order of discriminants, ” Acta Arithm., 133, 375–390 (2008).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Y. Bugeaud and A. Dujella, “Root separation for irreducible integer polynomials,” Bull. London Math. Soc., 162, 1239–1244 (2011).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Y. Bugeaud and A. Dujella, “Root separation for reducible integer polynomials,” Acta Arithm., 162, 393–403 (2014).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Y. Bugeaud and M. Mignotte, “On the distance between roots of integer polynomials,” Proc. Edinb. Math. Soc., 47, 553–556 (2004).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Y. Bugeaud and M. Mignotte, “Polynomial root separation,” Int. J. Number Theor., 6, 587–602 (2010).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    J.-H. Evertse, “Distances between the conjugates of an algebraic number,” Publ. Math. Debrecen, 65, 323–340 (2004).MathSciNetMATHGoogle Scholar
  10. 10.
    F. Götze, D. Kaliada, and M. Korolev, “On the number of integral quadratic polynomials with bounded heights and discriminants,” Preprint, arXiv:1308.2091 (2013).Google Scholar
  11. 11.
    D. Kaliada, F. Götze, and O. Kukso, “The asymptotic number of integral cubic polynomials with bounded heights and discriminants,” Preprint, arXiv:1307.3983 (2013).Google Scholar
  12. 12.
    K. Mahler, “An inequality for the discriminant of a polynomial,” Mich. Math. J., 11, 257–262 (1964).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    M. Mignotte, “Some useful bounds,” in: B. Buchberger et al. (eds), Computer Algebra. Symbolic and Algebraic Computation, Springer (1983), pp. 259–263.Google Scholar
  14. 14.
    B. L. van der Waerden, “Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt,” Monatsh. Mathematik, 43, 133–147 (1936).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Bielefeld UniversityBielefeldGermany
  2. 2.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

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