Large transcendence degrees for the exponential function; modification of the technical hypotheses in Brownawell's method
Let \(u_{1},\ldots,u_{n}\) be \(\mathbb{Q}\)-linearly independent complex numbers and \(v_{1},\ldots,v_{m}\) be also \(\mathbb{Q}\)-linearly independent complex numbers. Denote by \(t\) the transcendence degree over \(\mathbb{Q}\) of the field generated by the \(mn\) numbers \(e^{u_{i}v_{j}}\), (\(1\leq i\leq n\), \(1\leq j\leq m\)). It is well known (six exponentials theorem) that for \(mn>m+n\), we have \(t\geq 1\). Moreover, for \(mn\geq 2(m+n)\), we have \(t\geq 2\). Such results are called ``small transcendence degree''. Assume now \(mn>m+n\). After the work of several mathematicians, including W. D. Brownawell, G. V. Chudnovsky, E. Reyssat, P. Philippon and Yu. V. Nesterenko, G. Diaz established in 1986 the lower bound \(t\geq[mn/(m+n)]\) (``large transcendence degree''), subject to a so-called ``technical assumption'', namely a measure of linear independence for the numbers \(u_{1},\ldots,u_{n}\), and also for the numbers \(v_{1},\ldots,v_{m}\) [ \textit{G. Diaz}, J. Number Theory 31, 1-23 (1989; Zbl 0661.10047)]. The central open problem which is studied in these two papers under review (see Zbl 0878.11030) is to provide a lower bound for \(t\) in terms of \(mn/(m+n)\) without assuming such a technical assumption, or at least under weaker assumptions. In the first part, the author develops Diaz' method and proves two results. In the first one, she assumes only a technical hypothesis for a subset of \(u_{1},\ldots,u_{n}\), and also for a subset of \(v_{1},\ldots,v_{m}\). In the second one, she assumes the opposite of the usual technical hypothesis: she deals with the case where there is a sequence of linear forms which take very small values at \((u_{1},\ldots,u_{n})\). In the second part, she develops a method of \textit{W. D. Brownawell} [Lect. Notes Math. 1290, 149-173 (1987; Zbl 0626.10030)] who replaces Philippon's transcendence criterion by a Liouville-type inequality.
scientific article; zbMATH DE number 1013764
scientific article