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Multiplicative relations in number fields

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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In this paper, we obtain an explicit form of the currently best known inequality for linear forms in the logarithms of algebraic numbers. The results complete our previous investigations (Bull. Austral. Math. Soc. 15 (1976), 33–57) which were conditional on a certain independence condition on the algebraic numbers. The extra work needed to obtain unconditional results centres on the properties of multiplicative relations in number fields. In particular, we show that a set of multiplicatively dependent algebraic numbers always satisfies a relation with relatively small exponents.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 16 , Issue 1 , February 1977 , pp. 83 - 98
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Baker, A., “Linear forms in the logarithms of algebraic numbers (IV)”, Mathematika 15 (1968), 204–216.CrossRefGoogle Scholar
[2]Baker, A. and Stark, H.M., “On a fundamental inequality in number theory”, Ann. of Math. (2) 94 (1971), 190–199.CrossRefGoogle Scholar
[3]Blanksby, P.E. and Montgomery, H.L., “Algebraic integers near the unit circle”, Acta Arith. 18 (1971), 355–369.CrossRefGoogle Scholar
[4]Cassels, J.W.S., An introduction to the geometry of numbers, Second printing, corrected (Die Grundlehren der mathematischen Wissenschaften, 99. Springer-Verlag, Berlin, Heidelberg, New York, 1971).Google Scholar
[5]*Kronecker, L., “Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten”, J. reine angew. Math. 53 (1857), 173–175.Google Scholar
[6]*Landau, E., “Sur quelques théorèmes de M. Pétrovitch relatifs aux zéros des fonctions analytiques”, Bull. Soc. Math. France 33 (1905), 251–261.Google Scholar
[7]Ostrowski, A.M., “On an inequality of J. Vicente Gonçalves”, Univ. Lisboa Revista Fac. Ci. A (2) 8 (1960), 115–119.Google Scholar
[8]van der Poorten, A.J. and Loxton, J.H., “Computing the effectively computable bound in Baker's inequality for linear forms in logarithms”, Bull. Austral. Math. Soc. 15 (1976), 33–57.CrossRefGoogle Scholar
[9]Rosser, J. Barkley and Schoenfeld, Lowell, “Approximate formulas for some functions of prime numbers”, Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
[10]Schinzel, A. and Zassenhaus, H., “A refinement of two theorems of Kronecker”, Michigan Math. J. 12 (1965), 81–85.CrossRefGoogle Scholar
[11]Stark, H.M., “Further advances in the theory of linear forms in logarithms”, Diophantine approximation and its applications, 255–293 (Proc. Conf. Diophantine Approximation and its Applications, Washington, 1972. Academic Press [Harcourt-Brace Jovanovich], New York, London, 1973).Google Scholar