Conjugate algebraic numbers close to a symmetric set. (English. Russian original) Zbl 1092.11040
St. Petersbg. Math. J. 16, No. 6, 1013-1016 (2005); translation from Algebra Anal. 16, No. 6, 123-127 (2004).
It has been shown by Th. Motzkin [Bull. Am. Math. Soc. 53, 156–162 (1947; Zbl 0032.24702)] that if \(\{z_1,\dots, z_{n-1}\}\) is a set of complex numbers, closed under conjugation, then for every \(\epsilon>0\) there exists a monic polynomial \(P(X)\) of degree \(n\), with rational integral coefficients, irreducible over the rationals, such that if its roots \(w_1,\dots,w_n\) are suitably arranged, then for \(1\leq i\leq n-1\) one has \(| z_i-w_i| \leq\epsilon\). The author presents a new proof of this result, leading to an explicit construction of \(P(X)\).
Reviewer: Władysław Narkiewicz (Wrocław)
MSC:
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11D75 | Diophantine inequalities |
| 11J25 | Diophantine inequalities |
| 12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
Citations:
Zbl 0032.24702References:
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