Linear Forms in the Periods of the Exponential and Elliptic Functions.
1. Introduction. Our aim is to test numerically the new method of interpolation determinants (cf. [2], [6]) in the context of linear forms in two logarithms. In the recent years, M. Mignotte and M. Waldschmidt have used Schneider's construction in a series of papers [3]-[5] to get lower bounds for such a linear form with rational integer coefficients. They got relatively precise results with a numerical constant around a few hundreds. Here we take up Schneider's method again in the framework...
We prove a result on the existence of linear forms of a given Diophantine type.
Let be a real number with continued fraction expansionand letbe a matrix with integer entries and nonzero determinant. If has bounded partial quotients, then also has bounded partial quotients. More precisely, if for all sufficiently large , then for all sufficiently large . We also give a weaker bound valid for all with . The proofs use the homogeneous Diophantine approximation constant . We show that
The main result of this paper is a criterion for linear independence of continued fractions over the rational numbers. The proof is based on their special properties.
For , , , let be the -th polylogarithm of . We prove that for any non-zero algebraic number such that , the -vector space spanned by has infinite dimension. This result extends a previous one by Rivoal for rational . The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.
Let be a nonconstant polynomial with integer coefficients and without zeros at the non–negative integers. Essentially with the method of Hermite, a new proof is given on linear independence of values at rational points of the function