# zbMATH — the first resource for mathematics

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
##### Keywords:
algebraic number; cluster; integer polynomial; height
Full Text:
##### References:
 [1] A. Baker, W. M. Schmidt, Proc. London Math. Soc. (3), 21 (1970), 1 – 11 · Zbl 0206.05801 · doi:10.1112/plms/s3-21.1.1 [2] В. Г. Спринджук, Проблема Малера в метрической теории чисел, Наука и техника, Минск, 1967 · Zbl 0168.29504 [3] K. Mahler, Michigan Math. J., 11:3 (1964), 257 – 262 · Zbl 0135.01702 · doi:10.1307/mmj/1028999140 [4] V. V. Beresnevich, V. I. Bernik, F. Goẗze, Compositio Math., 146:5 (2010), 1165 – 1179 · Zbl 1206.11091 · doi:10.1112/S0010437X10004860 · arxiv:0906.4286 [5] Y. Bugeaud, M. Mignotte, Proc. Edinb. Math. Soc. (2), 47:2 (2004), 553 – 556 · Zbl 1071.11016 · doi:10.1017/S0013091503000257 [6] Y. Bugeaud, M. Mignotte, Int. J. Number Theory, 6:3 (2010), 587 – 602 · Zbl 1205.11032 · doi:10.1142/S1793042110003083 [7] Y. Bugeaud, A. Dujella, Bull. Lond. Math. Soc., 43:6 (2011), 1239 – 1244 · Zbl 1273.11049 · doi:10.1112/blms/bdr085 [8] Y. Bugeaud, A. Dujella, Root Separation for Reducible Integer Polynomials, arXiv: math.NT/1306.2128v1 [9] В. В. Бересневич, В. И. Берник, Ф. Гетце, Докл. НАН Беларуси, 54:5 (2010), 21 – 23 · Zbl 1267.11026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.