Algebraische Unabhängigkeit der Werte gewisser Lückenreihen in nicht-archimedisch bewerteten Körpern

Abstract

In 1844 Liouville proved the transcendence of α = ∑h≥1 10−h h! over Q. The number α can be considered as the value of the gap power series ∧(x) =∑h≥1 at tne point 1/10 Since then, the above result has been generalized in this direction by different authors by applying improved “Liouville-estimates”. For instance, in 1973 Cijsouw and Tijdeman [2] showed that a gap series with algebraic coefficients takes on transcendental values (over Q) at non-zero algebraic points under some conditions on the growth of the coefficients and the gaps. In 1988 Bundschuh [1] resp. Zhu [9] proved the algebraic independence (over Q) of the values of several gap series at different algebraic points. In particular this result includes the algebraic independence of A(α1),…, α(αs) for non-zero algebraic numbers α1,…, αs of distinct absolute values less than 1. Moreover in [1] a set of continuum-many algebraically independent numbers was constructed.