Skip to main content

Algebraic points on the plane


This article gives quantitative estimates for the measure of points (x, y) of a given rectangle admitting the construction of polynomials P(t) with small (with respect to the height of the polynomial) values of P(x) and P(y). Such estimates can be used in the problem of distribution of algebraic points on the plane.

This is a preview of subscription content, access via your institution.


  1. 1.

    V. V. Beresnevich, “On approximation of real numbers by real algebraic integers,” Acta Arith., 90, No. 2, 97–112 (1999).

    MATH  MathSciNet  Google Scholar 

  2. 2.

    V. V. Beresnevich, “A Groshev type theorem for convergence on manifolds,” Acta Math. Acad. Sci. Hungar., 94, Nos. 1–2, 99–130 (2002).

    MATH  MathSciNet  Google Scholar 

  3. 3.

    V. V. Beresnevich, “Distribution of rational points near parabola,” Dokl. Akad. Nauk Belarusi, 46, 13–15 (2002).

    MathSciNet  Google Scholar 

  4. 4.

    V. V. Beresnevich, V. Bernik, D. Kleinbock, and G. Margulis, “Metric Diophantine approximation: The Khinchine-Groshev type theorem for nondegenerable manifolds,” Moscow Math. J., 2, No. 2, 203–225 (2000).

    MathSciNet  Google Scholar 

  5. 5.

    V. I. Bernik, “A metric theorem on the simultaneous approximation of zero by the values of integral polynomials,” Math. USSR-Izv., 16, No. 1, 21–40 (1981).

    MATH  Article  Google Scholar 

  6. 6.

    V. I. Bernik, “On the exact order of approximation of zero by values of integral polynomials,” Acta Arith., 53, No. 1, 17–28 (1989).

    MATH  MathSciNet  Google Scholar 

  7. 7.

    V. I. Bernik and V. N. Borbat, “Simultaneous approximation of zero by values of integral polynomials,” Proc. Steklov Inst. Math., 218, 53–68 (1997).

    MathSciNet  Google Scholar 

  8. 8.

    V. Bernik, D. Kleinbock, and G. Margulis, “Khinchine-type theorem on manifolds: The convergence case and multiplicative versions,” Intern. Math. Research Notes, No. 9, 453–486 (2001).

  9. 9.

    M. Huxley, Area, Lattice Points and Exponential Sums, Oxford (1996).

  10. 10.

    N. A. Pereverseva, “Simultaneous approximation of zero by values of relatively prime integral polynomials,” Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. (1984).

  11. 11.

    V. G. Sprindzuk, Mahler’s Problem in Metric Number Theory, Amer. Math. Soc., Providence (1969).

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to V. V. Lebed.

Additional information


Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 73–80, 2005.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lebed, V.V., Bernik, V.I. Algebraic points on the plane. J Math Sci 146, 5680–5685 (2007).

Download citation


  • Manifold
  • Simultaneous Approximation
  • Diophantine Approximation
  • Algebraic Point
  • Reducible Polynomial