An axiomatization of Nesterenko's method and applications on Mahler functions (Q1336053)

The author proves a new criterion for algebraic independence of complex numbers, using Nesterenko's method based on Cayley-Chow forms [\textit{Yu. V. Nesterenko}, Math. USSR, Izv. 11, 239-270 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253-284 (1977; Zbl 0354.10026)]. The main difference with Philippon's earlier criterion [\textit{P. Philippon}, Publ. Math., Inst. Hautes Etud. Sci. 64, 5-52 (1986; Zbl 0615.10044)] is that here, only a finite sequence of polynomials is required [see also \textit{E. M. Jabbouri}, Approximations diophantiennes et nombres transcendants, C.-R. Colloq., Luminy/Fr. 1990, de Gruyter, 195- 202 (1992; Zbl 0777.11024)]. Several applications are provided. The first one is a generalization to several variables of Nishioka's results on algebraic independence of values of Mahler's functions [\textit{K. Nishioka}, J. Reine Angew. Math. 407, 202-219 (1990; Zbl 0694.10035)]. The second one yields algebraic independence for values at Liouville points of functions satisfying functional equations. The third one, also connected with Mahler's method, is an extension of a result of \textit{M. Amou} [Acta Arith. 59, 71-82 (1991; Zbl 0735.11031)]. Finally the author applies his criterion to the usual exponential function, but his result is weaker than the one of \textit{G. Diaz} [J. Number Theory 31, 1-23 (1989; Zbl 0661.10047)].