Far Eastern Mathematical Journal

To content of the issue


On the distribution of real algebraic numbers of equal height


Koleda D.V.

2018, issue 1, P. 56-70


Abstract
In the paper we find the asymptotic number of algebraic numbers of fixed degree $n\ge 1$ and height H lying in an interval $I\subseteq\mathbb{R}$ as $H\to\infty$.

Keywords:
algebraic numbers, distribution of algebraic numbers, integer polynomials, generalized Farey sequences

Download the article (PDF-file)

References

[1] S.-J. Chern, J.D. Vaaler, “The distribution of values of Mahler’s measure”, J. Reine Angew. Math., 540, (2001), 1–47.
[2] D. Masser, J.D. Vaaler, “Counting algebraic numbers with large height I”, Diophantine Approximation, Dev. Math., 16, Springer, Vienna, 2008, 237–243.
[3] D. Masser, J.D. Vaaler, “Counting algebraic numbers with large height. II”, Trans. Amer. Math. Soc., 359:1, (2007), 427–445.
[4] R. Grizzard, J. Gunther, “Slicing the stars: counting algebraic numbers, integers, and units by degree and height”, Algebra Number Theory, 11:6, (2017), 1385–1436.
[5] H. Brown, K. Mahler, “A generalization of Farey sequences: Some exploration via the computer”, J. Number Theory, 3:3, (1971), 364–370.
[6] D.U. Kaljada, “Ab razmerkavanni rjechaisnyh algebraichnyh likaў dadzenaj stupeni”, Doklady NAN Belarusi, 56:3, (2012), 28–33.
[7] D.V. Koleda, “O raspredelenii dejstvitel'nyh algebraicheskih chisel vtoroj stepeni”, Vesci NAN Belarusi. Ser. fiz.-mat. navuk, 2013, № 3, 54–63.
[8] D. Koleda, “On the density function of the distribution of real algebraic numbers”, J. Theor. Nombres Bordeaux, 29:1, (2017), 179–200.
[9] D.N. Zaporozhec, “Sluchajnye polinomy i geometricheskaja verojatnost'”, Doklady Akademii nauk, 400:3, (2005), 299–303.
[10] F. Gotze, D.V. Koleda, D.N. Zaporozhets, “Correlations between real conjugate algebraic numbers”, Chebyshevskii Sb., 16:4, (2015), 90–99.
[11] A. Dubickas, “On the number of reducible polynomials of bounded naive height”, Manuscripta Mathematica, 144:3–4, (2014), 439–456.
[12] H. Davenport, “On a principle of Lipschitz”, J. Lond. Math. Soc., 26:3, (1951), 179–183.
[13] K. Prahar, Raspredelenie prostyh chisel, Mir, M., 1967.
[14] V.V. Prasolov, Mnogochleny, 3-e izd., ispr., MCNMO, M., 2003.

To content of the issue