Ideal constructions and irrationality measures of roots of algebraic numbers. (Q1850225)

The authors study the possible limits of the Thue-Siegel method, as developed by E. Bombieri to obtain effective irrationality measures to roots of high order of algebraic numbers. For simplicity they consider only non-archimedean valuations, the archimedean case should be treated in a similar way. They study in detail the construction of auxiliary functions and use a version of the Thue-Siegel lemma due to \textit{T. Struppeck} and \textit{J. Vaaler} [Prog. Math. 85, 493--528 (1990; Zbl 0722.11033)]. Instead of using some version of Dyson's lemma they just assume some non-vanishing condition for their auxiliary functions. Under this (strong) hypothesis they get results comparable to the best ones obtained by transcendental methods.