forms in to crystals

Linear forms in logarithms

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

Σ βi log αi = Λ

where the αi and βi are algebraic numbers. In case of αi 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 αi and βi. 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.h1)h2) log B

with a constant C

C = 18(n+1)! nn+1 (32d)n+2log(2n d)

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

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