# Effective simultaneous approximation of complex numbers by conjugate algebraic integers

Acta Arithmetica (1993)

- Volume: 63, Issue: 4, page 325-334
- ISSN: 0065-1036

## Access Full Article

top## Abstract

top## How to cite

topG. J. Rieger. "Effective simultaneous approximation of complex numbers by conjugate algebraic integers." Acta Arithmetica 63.4 (1993): 325-334. <http://eudml.org/doc/206524>.

@article{G1993,

abstract = {
We study effectively the simultaneous approximation of n-1 different complex numbers by conjugate algebraic integers of degree n over ℤ(√-1). This is a refinement of a result of Motzkin [2] (see also [3], p. 50) who has no estimate for the remaining conjugate. If the n-1 different complex numbers lie symmetrically about the real axis, then ℤ(√-1) can be replaced by ℤ.
In Section 1 we prove an effective version of a Kronecker approximation theorem; we start with an idea of H. Bohr and E. Landau (see e.g. [4]); later we use an estimate of A. Baker for linear forms with logarithms. This and also Rouché's theorem are then applied in Section 2 to give the result; the required irreducibility is guaranteed by the Schönemann-Eisenstein criterion.
},

author = {G. J. Rieger},

journal = {Acta Arithmetica},

keywords = {Baker's method; Kronecker theorem},

language = {eng},

number = {4},

pages = {325-334},

title = {Effective simultaneous approximation of complex numbers by conjugate algebraic integers},

url = {http://eudml.org/doc/206524},

volume = {63},

year = {1993},

}

TY - JOUR

AU - G. J. Rieger

TI - Effective simultaneous approximation of complex numbers by conjugate algebraic integers

JO - Acta Arithmetica

PY - 1993

VL - 63

IS - 4

SP - 325

EP - 334

AB -
We study effectively the simultaneous approximation of n-1 different complex numbers by conjugate algebraic integers of degree n over ℤ(√-1). This is a refinement of a result of Motzkin [2] (see also [3], p. 50) who has no estimate for the remaining conjugate. If the n-1 different complex numbers lie symmetrically about the real axis, then ℤ(√-1) can be replaced by ℤ.
In Section 1 we prove an effective version of a Kronecker approximation theorem; we start with an idea of H. Bohr and E. Landau (see e.g. [4]); later we use an estimate of A. Baker for linear forms with logarithms. This and also Rouché's theorem are then applied in Section 2 to give the result; the required irreducibility is guaranteed by the Schönemann-Eisenstein criterion.

LA - eng

KW - Baker's method; Kronecker theorem

UR - http://eudml.org/doc/206524

ER -

## References

top- [1] A. Baker, Transcendental Number Theory, Cambridge Univ. Press, 1975. Zbl0297.10013
- [2] T. Motzkin, From among n conjugate algebraic integers, n-1 can be approximately given, Bull. Amer. Math. Soc. 53 (1947), 156-162. Zbl0032.24702
- [3] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN, Warszawa 1974. Zbl0276.12002
- [4] P. Turán, Nachtrag zu meiner Abhandlung 'On some approximative Dirichlet polynomials in the theory of zeta-function of Riemann', Acta Math. Acad. Sci. Hungar. 10 (1959), 277-298. Zbl0103.04503

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.