Effective simultaneous approximation of complex numbers by conjugate algebraic integers
Acta Arithmetica, Tome 63 (1993) no. 4, pp. 325-334
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.
@article{10_4064_aa_63_4_325_334,
author = {G. Rieger},
title = {Effective simultaneous approximation of complex numbers by conjugate algebraic integers},
journal = {Acta Arithmetica},
pages = {325--334},
year = {1993},
volume = {63},
number = {4},
doi = {10.4064/aa-63-4-325-334},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa-63-4-325-334/}
}
TY - JOUR AU - G. Rieger TI - Effective simultaneous approximation of complex numbers by conjugate algebraic integers JO - Acta Arithmetica PY - 1993 SP - 325 EP - 334 VL - 63 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4064/aa-63-4-325-334/ DO - 10.4064/aa-63-4-325-334 LA - en ID - 10_4064_aa_63_4_325_334 ER -
G. Rieger. Effective simultaneous approximation of complex numbers by conjugate algebraic integers. Acta Arithmetica, Tome 63 (1993) no. 4, pp. 325-334. doi: 10.4064/aa-63-4-325-334
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