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

Published online by Cambridge University Press:  17 April 2009

A.J. van der Poorten  and
J.H. Loxton
A.J. van der Poorten
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
J.H. Loxton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
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Abstract

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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)

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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 1 , August 1976 , pp. 33 - 57
Copyright
Copyright © Australian Mathematical Society 1976

References

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