Distance between conjugate algebraic numbers in clusters

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, …, ξt by conjugate algebraic numbers of bounded degree over ℚ, provided that the given transcendental numbers ξ1, …, ξt generate over ℚ a field of transcendence degree one. We provide sharper estimates for example when ξ1, …, ξt form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at ξ1, …, ξt.

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ON SUBFIELDS OF A FIELD GENERATED BY TWO CONJUGATE ALGEBRAIC NUMBERS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091503000427 ◽

2004 ◽

Vol 47 (1) ◽

pp. 119-123 ◽

Cited By ~ 6

Author(s):

Paulius Drungilas ◽

Artūras Dubickas

Keyword(s):

Algebraic Number ◽

Monic Polynomial ◽

Subject Classification ◽

Real Field ◽

Algebraic Numbers ◽

Root Of Unity ◽

Conjugate Algebraic Numbers

AbstractLet $k$ be a field, and let $\alpha$ and $\alpha'$ be two algebraic numbers conjugate over $k$. We prove a result which implies that if $L\subset k(\alpha,\alpha')$ is an abelian or Hamiltonian extension of $k$, then $[L:k]\leq[k(\alpha):k]$. This is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in $X$—irreducible over a real field and having $m$ roots of equal modulus, at least one of which is real—is a polynomial in $X^m$.AMS 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10

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Conjugate algebraic numbers on circles

Acta Arithmetica ◽

10.4064/aa-29-2-147-157 ◽

1976 ◽

Vol 29 (2) ◽

pp. 147-157

Author(s):

Veikko Ennola ◽

C. Smyth

Keyword(s):

Algebraic Numbers ◽

Conjugate Algebraic Numbers

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Multiplicative relations with conjugate algebraic numbers

Ukrainian Mathematical Journal ◽

10.1007/s11253-007-0064-6 ◽

2007 ◽

Vol 59 (7) ◽

pp. 984-995

Author(s):

A. Dubickas

Keyword(s):

Algebraic Numbers ◽

Conjugate Algebraic Numbers

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The distribution of close conjugate algebraic numbers

Compositio Mathematica ◽

10.1112/s0010437x10004860 ◽

2010 ◽

Vol 146 (5) ◽

pp. 1165-1179 ◽

Cited By ~ 15

Author(s):

Victor Beresnevich ◽

Vasili Bernik ◽

Friedrich Götze

Keyword(s):

Diophantine Approximation ◽

Real Line ◽

Algebraic Numbers ◽

Fixed Degree ◽

Irreducible Polynomials ◽

The Real ◽

Conjugate Algebraic Numbers

AbstractWe investigate the distribution of real algebraic numbers of a fixed degree that have a close conjugate number, with the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general, it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds obtained by Bugeaud and Mignotte. So far, the results à la Bugeaud and Mignotte have relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. We consider some applications of our main theorem, including generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.

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