Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

A. J. Van Der Poorten; J. H. Loxton

doi:10.1017/S0004972700036741

Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

A. J. Van Der Poorten, J. H. Loxton

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α₁, …, α_n all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n₀(D) which is explicitly computed, the inequalities (in the standard notation) [formula omitted] have no solution in rational integers b₁, …, b_n (b_n ≠ 0) of absolute value at most B, whenever n ≥ n₀(D). The very favourable dependence on n is particularly useful.

Original language	English
Pages (from-to)	33-57
Number of pages	25
Journal	Bulletin of the Australian Mathematical Society
Volume	15
Issue number	1
DOIs	https://doi.org/10.1017/S0004972700036741
Publication status	Published - 1976
Externally published	Yes

Bibliographical note

Corrigenda and addenda can be found in Bulletin of the Australian Mathematical Society, 17(1), pp. 151-155, 1977.
https://doi.org/10.1017/S0004972700025600

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10.1017/S0004972700036741

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abstract = "For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation) [formula omitted] have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.",

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AU - Loxton, J. H.

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N2 - For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation) [formula omitted] have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

AB - For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation) [formula omitted] have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

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Computing the effectively computable bound in Baker's inequality for linear forms in logarithms

Abstract

Bibliographical note

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