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In number theory the method of linear forms in logarithms is the application of estimates
for the magnitude of a finite sum## See also

## References

- Σ β
_{i}log α_{i}= Λ

where the α_{i} and β_{i} are algebraic numbers. In case of α_{i} a complex number, one has to allow log to denote some definite branch of the logarithm function in the complex plane. Applications include transcendence theory, establishing measures of transcendence of real number, and the effective resolution of Diophantine equations. It has been suitably generalised to elliptic logarithms and functions on abelian varieties.

The class of results established by Alan Baker's work supply lower bounds for |Λ|, in cases where Λ ≠ 0. This is in terms of quantities A and B, respectively bounding the heights of the α_{i} and β_{i}. This work supplied many results on diophantine equations, amongst other applications.

A recent explicit result by Baker and Wüstholz for a linear form |Λ| with integer coefficients yields a lower bound of the form

- log |Λ| > -C.h(α
_{1})h(α_{2})...h(α_{n}) log B

with a constant C

- C = 18(n+1)! n
^{n+1}(32d)^{n+2}log(2n d)

where d is the degree of the number field generated by the αs.

- A. Baker and G. Wustholz, "Logarithmic forms and group varieties", J. Reine Angew. Math. 442 (1993) 19-62
- N. Smart, "The algorithmic resolution of Diophantine equations", Cambridge University Press, 1998, ISBN 0-521-64156-X. App.A.

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Last updated on Saturday June 14, 2008 at 13:05:14 PDT (GMT -0700)

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