On some applications of diophantine approximations. (English) Zbl 0544.10034

Chudnovsky, G. V.

Proc. Natl. Acad. Sci. USA 81, 1926-1930 (1984).

The author greatly improves on previous results on general E-functions by showing that if the number field appearing in the definition of E- functions is \({\mathbb{Q}}\), a measure of linear independence over \({\mathbb{Q}}\) for their values at rational points can be obtained with a Dirichlet exponent depending solely on their number (rather than on the order of the differential system they are assumed to satisfy). The main new tool introduced in the proof is the construction of a so-called set of graded Padé approximations, whose normality properties have already been studied by the author and D. V. Chudnovsky [Lect. Notes Math. 1052, 85-167 (1984; Zbl 0536.10029)].

Reviewer: D.Bertrand

MSC:

11J81	Transcendence (general theory)
41A21	Padé approximation
34A40	Differential inequalities involving functions of a single real variable
33C60	Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)

Keywords:

irrationality measures; E-functions; measure of linear independence; values at rational points; graded Padé approximations

Citations:

Zbl 0536.10029