Simultaneous approximation and algebraic independence

被引：0

作者：

Roy D. ^{[1
]}

Waldschmidt M. ^{[2
]}

机构：

[1] Department de Mathématiques, Université d'Ottawa, Ottawa, Ont.

[2] Institut de Mathématiques de Jussieu, Problèmes Diophantiens, 75252 Paris Cedex 05, 4, Place Jussieu

来源：

The Ramanujan Journal | 1997年 / 1卷 / 4期

基金：

加拿大自然科学与工程研究理事会;

关键词：

Algebraic independence; Approximation measures; Diophantine estimates; Dirichlet's box principle; Gel'fond's criterion; Interpolation determinants; Liouville's inequality; Simultaneous approximation; Transcendental numbers; Wirsing's theorem;

D O I：

10.1023/A:1009757810489

中图分类号：

学科分类号：

摘要：

We establish several new measures of simultaneous algebraic approximations for families of complex numbers (θ1, ..., θn) related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the field ℚ(θ1, ..., θn) has transcendance degree ≥2 over ℚ. This approach which is ultimately based on the technique of interpolation determinants provides an alternative to Gel'fond's transcendence criterion. We also formulate a conjecture about simultaneous algebraic approximation which would yield higher transcendance degrees from these measures. © 1997 Kluwer Academic Publishers.

引用

页码：379 / 430

页数：51

共 9 条

[1]

Baker A., Transcendental Number Theory, 2nd Edition, (1979)

[2]

Barre-Sirieix K., Diaz G., Gramain F., Philibert G., Une preuve de la conjecture de Mahler-Manin, Invent. Math., 124, pp. 1-9, (1996)

[3]

Bertrand D., Philippon P., Sous-groupes algébriques de groupes algébriques commutatifs, Illinois J. Math., 32, pp. 263-280, (1988)

[4]

Brownawell W.D., The algebraic independence of the values of the exponential function, K. Norske Vidensk. Selsk. Skr., 23, (1972)

[5]

Brownawell W.D., Sequences of diophantine approximations, J. Number Theory, 6, pp. 11-21, (1974)

[6]

Chudnovsky G.V., Contributions to the Theory of Transcendental Numbers, Math. Surveys and Monographs, 19, (1984)

[7]

Chudnovsky G.V., Algebraic independence of the values of elliptic functions at algebraic points, Invent. Math., 61, pp. 267-290, (1980)

[8]

Diaz G., Une nouvelle minoration de| log α - β|, |α - exp

[9]

β|, α et β algébriques, Acta Arith., 64, pp. 43-57, (1993)

← 1 →