Quantitative results on algebraic independence for algebraic groups
The results of algebraic independence, due to Gel'fond, Chudnovsky and others, concerning the values \(\exp(x_iy_j)\) of the exponential function (for complex numbers \(x_1,\ldots x_n\) and \(y_1,\ldots,y_m\)) have been improved by \textit{G. Diaz} [ J. Number Theory 31, 1--23 (1989; Zbl 0661.10047)]. Here, the author extends these theorems of Diaz on large transcendence degree to one parameter subgroups of commutative algebraic groups. At the same time he provides quantitative versions (measures of algebraic independence). His main theorem contains a lot of corollaries; see for instance the author's note reviewed below [C. R. Acad. Sci., Paris, Sér. I 313, No. 10, 653--655 (1991; Zbl 0758.11033)]. One of his main tools is the use of Chow forms introduced by Nesterenko and developed by Philippon (see also \textit{E. M. Jabbouri} [Sur un critère pour l'indépendance algébrique de P. Philippon, Approximations diophantiennes et nombres transcendants, Luminy 1990, 195--202 (1992; Zbl 0777.11024)]).
one-parameter subgroups of commutative algebraic groups
scientific article; zbMATH DE number 93289
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