On the distribution of algebraic numbers having a given degree over a number field

In the paper we consider the question: given a number field K, what is the distribution of algebraic numbers α of fixed degree deg Kα = n and height H Q ( ) α ≤ in R and in C as Q → ∞. We establish the asymptotic count of such algebraic numbers lying in a set S ⊂ C in cases when S is an interval of the real line or a region of the complex plane with a Lipschitz-parametrizable boundary. In particular, we show that the asymptotic distribution density function of such algebraic numbers is completely determined by their degree over K, by the internal structure of the height function H and by what the completion of K with respect to the archimedean metric is (whether it is R or C ).

1. Lang S. Diophantine Geometry. Interscience Tracts in Pure and Applied Mathematics, vol. 11. New York, London, 1962. 170 p.

2. Schanuel S. H. Heights in number fields. Bulletin de la Société Mathématique de France, 1979, vol. 107, pp. 433–449. https://doi.org/10.24033/bsmf.1905

3. Schmidt W. M. Northcott’s theorem on heights I. A general estimate. Monatshefte für Mathematik, 1993, vol. 115, pp. 169–181. https://doi.org/10.1007/BF01311215

4. Masser D., Vaaler J. D. Counting algebraic numbers with large height II. Transactions of the American Mathematical Society, 2007, vol. 359, no. 1, pp. 427–445. https://doi.org/10.1090/S0002-9947-06-04115-8

5. Widmer M. Counting primitive points of bounded height. Transactions of the American Mathematical Society, 2010, vol. 362, pp. 4793–4829. https://doi.org/10.1090/S0002-9947-10-05173-1

6. Widmer M. Integral points of fixed degree and bounded height. International Mathematics Research Notices, 2016, vol. 2016, no. 13, pp. 3906–3943. https://doi.org/10.1093/imrn/rnv268

7. Koleda D. On the density function of the distribution of real algebraic numbers. Journal de Théorie des Nombres de Bordeaux, 2017, vol. 29, no. 1, pp. 179–200. https://doi.org/10.5802/jtnb.975

8. Götze F., Kaliada D., Zaporozhets D. Distribution of complex algebraic numbers. Proceedings of the American Mathematical Society, 2017, vol. 145, no. 1, pp. 61–71. https://doi.org/10.1090/proc/13208

9. Götze F., Koleda D., Zaporozhets D. Joint distribution of conjugate algebraic numbers: a random polynomial approach. Advances in Mathematics, 2020, vol. 359, art. 106849. https://doi.org/10.1016/j.aim.2019.106849

10. Shabat B. V. Introduction to Complex Analysis. Part II. Functions of several variables, second ed. Moscow, 1976. 400 p. (in Russian).