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General mathematics seminar

October 18, 2021. D. N. Zaporozhets. On distribution of algebraic numbers.

It follows from the group structure of rational points on the unit circle that in some sense they are uniformly distributed. The well-known "rational parametrization" ρ : ℝ1 → 𝕊1 of the unit circle

which is nothing but the inverse stereographic projection from the upper point of the circle onto the abscissa, sets a one-to-one correspondence between the rational points on the unit circle (except the upper one) and all rational numbers. In particular, it gives one way of finding all Pythagorean triples.

The stereographic projection maps the uniform measure on the circle to the measure on the line with the Cauchy density . Thus we obtain that, in some sense, the rational numbers are distributed with respect to this density (the rigorous formulation will be given).

The rational numbers are the algebraic numbers of degree 1. In this talk, we aim to generalize the above elementary observation with the Cauchy density to the algebraic numbers of arbitrary degree n. To this end, we will apply the theory of zeros of random polynomials.

Based on the joint paper

F. Götze, D. Koleda, D. Zaporozhets, "Joint distribution of conjugate algebraic numbers: a random polynomial approach", Adv. Math., 359 (2020)

List of talks at previous sessions of the seminar.