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Distance between conjugate algebraic numbers in clusters. (English. Russian original) Zbl 1284.11051
Math. Notes 94, No. 5, 816-819 (2013); translation from Mat. Zametki 94, No. 5, 780-783 (2013).
Summary: For integers \(n \geq 2\) and \(Q > 1\), the following class of integer polynomials is defined
\[ P(x) = a_n x^n + a_{n-1}x^{n-1}+ \cdots + a_1 x + a_0 \]
such that
\[ \mathcal{P}_n(Q) = \{ P \in \mathbb{Z}[x]: \text{deg} \, \text{deg } P = n, H(P) \leq Q\} \] , where \(H=H(P) = \max_{0\leq j \leq n}|a_j|\) is the height of \(P\). Let \(\alpha_1 \dots \alpha_n \in \mathbb{C}\), \(\alpha_i \neq \alpha_j\) be the roots of \(P\).
A systematic study of the quantities \(|\alpha_i - \alpha_j|\) for various conjugates algebraic numbers \(\alpha_i\) and \(\alpha_j\) has been done and also of the more general problem of clusters
\[ M_k = \prod_{1 \leq i < j \leq k} |\alpha_i - \alpha_j| \]
Next, denote by \(E(n,k)\) (resp. \(E_{\text{irr}}(n,k))\) the infimum of real numbers \(\delta\) for which the inequality
\[ \prod_{1 \leq i < j \leq k} |\alpha_i -\alpha_j| \geq H(P)^{-\delta} \]
holds for each integer (resp. integer irreducible) polynomial \(P\) of degree \(n\). In the present paper the authors obtain sharp estimates for \(E_{\text{irr}}(n,k))\).

MSC:
11B83 Special sequences and polynomials
11J68 Approximation to algebraic numbers
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