Joint distribution of conjugate algebraic numbers: a random polynomial approach

Distribution of complex algebraic numbers

“…As was mentioned above, the results on the distribution of the roots of random algebraic polynomials were used in [7], [8] to study the asymptotic behavior of algebraic numbers in a domain of the complex plane. In this paper we show that the distribution of algebraic numbers on the unit circle can be described in terms of random trigonometric polynomials.…”

“…Later Koleda [9] determined the asymptotic number of real algebraic numbers ordered by the naïve height; the same result for complex algebraic numbers was obtained in [7]. A generalization of the naïve height, the weighted l p -norm (which also generalizes the length, the Euclidean norm, and the Bombieri p-norm), was considered in [8]. Another interesting example of heights -namely the house of an algebraic number -was studied in [2], which studies the distribution of Perron numbers.…”

“…Before we do it, we need to check that the boundary of A l is of Lipschitz class. This lemma is a slightly modified and simplified version of [8,Lemma 6.4]. For the reader's convenience, we give a detailed proof of Lemma 3.3 in Section 4.…”

On the distribution of Salem numbers

“…Problems concerning the distribution of algebraic numbers have a long history [26,19,20,23,22,21]. Recall that an algebraic number α is a complex number such that there exists an irreducible polynomial P over Q with integer co-prime coefficients and positive leading coefficient such that P (α) = 0.…”