Let n ≥ 1 be an integer and α1, . . . , αn be non-zero algebraic numbers. Let b1, . . . , bn be integers with bn ≠ 0, and set B = max{3, |b1|, . . . , |bn|}. For j = 1, . . . , n, set h*(αj) =max{h(αj),2}, where h denotes the (logarithmic) Weil height. Assume that the quantity Λ = b1 log α1 + · · · + bn log αn is non-zero. A typical lower bound of log |Λ| given by Baker’s theory of linear forms in logarithms takes the shape
–c(n,D)h*(α1) . . . h*(αn) log B,
where c(n,D) is positive, effectively computable and depends only on n and on the degree D of the field generated by α1, . . . , αn. However, in certain special cases and in particular when |bn| = 1, this bound can be improved to
–c(n,D)h*(α1) . . . h*(αn) log (B/h*(αn)).
The term B‘= B/h*(αn) in place of B originates in works of Feldman and Baker. It is a key tool for improving, in an effective way, the upper bound for the irrationality exponent of a real algebraic number of degree at least 3 given by Liouville’s theorem. We survey various applications of this B‘ to exponents of approximation evaluated at algebraic numbers, to the S-part of integer sequences, and to Diophantine equations.
This video is part of the Number Theory Web Seminar series.
