Effective simultaneous approximation of complex numbers by conjugate algebraic integers

• Autorzy: G. J. Rieger
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1993
• Tom: 63
• Numer: 4
• Strony: 325-334

Daty

wydano

1993

otrzymano

1992-02-21

poprawiono

1992-05-08

Twórcy

autor

G. J. Rieger

• Institut für Mathematik, Universität Hannover, D-3000 Hannover, Germany

Bibliografia

• [1] A. Baker, Transcendental Number Theory, Cambridge Univ. Press, 1975.
• [2] T. Motzkin, From among n conjugate algebraic integers, n-1 can be approximately given, Bull. Amer. Math. Soc. 53 (1947), 156-162.
• [3] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN, Warszawa 1974.
• [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.