Linear forms in logarithms
- Linear forms in logarithms
In number theory the method of linear forms in logarithms is the application of estimatesfor the magnitude of a finite sum
: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"(α1)"h"(α2)..."h"(α"n") log "B"
with a constant "C"
:"C" = 18("n"+1)! "n""n"+1 (32"d")"n"+2log(2"n d")
where "d" is the degree of the number field generated by the αs.
ee also
* Logarithmic form
References
* 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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