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Generalization of the criteria for linear independence of Nesterenko, Amoroso, Colmez, Fischler and Zudilin. (Généralisation des critères pour l’indépendance linéaire de Nesterenko, Amoroso, Colmez, Fischler et Zudilin.) (French. English summary) Zbl 1252.11056
A criterion for linear independence, due to Yu. V. Nesterenko plays a central role in many investigations concerning linear independence results and measures, including proofs of irrationality and irrationality measures. The initial proof by Nesterenko has been simplified by F. Amoroso and P. Colmez. Another approach has been introduced by S. Fischler and W. Zudilin. In the paper under review, the author pursues her investigation [A. Chantanasiri, On the criteria for linear independence of Nesterenko, Fischler and Zudilin, Chamchuri J. Math. 2, No 1, 31–46 (2010)], where she extended the method of Fischler and Zudilin and studied sequences of numbers in place of tuples of numbers, obtaining also criteria for transcendence and algebraic independence. Here, she develops both methods and compares carefully the results which can be derived. She studies the real, complex and ultrametric cases. Finally, she investigates the optimality of her results.

##### MSC:
 11J72 Irrationality; linear independence over a field
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##### References:
 [1] Amoroso, F., Independenza lineare [2] Chantanasiri, A., On the criteria for linear independence of nesterenko, fischler and zudilin, Chamchuri Journal of Mathematics, 2, 1, 31-46, (2010) · Zbl 1282.11080 [3] Colmez, P., La fonction zêta, Arithmétique de la fonction zêta, 37-164, (2003), Ed. Éc. Polytech., Palaiseau [4] Fischler, S.; Zudilin, W., A refinement of nesterenko’s linear independence criterion with applications to zeta values, Math. Ann., 347, 4, 739-763, (2010) · Zbl 1206.11088 [5] Nesterenko, Yu. V., Linear independence of numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1, 108, 46-49, (1985) · Zbl 0572.10027 [6] Nesterenko, Yu. V., On a criterion of linear independence of $$p$$-adic numbers, (2011) · Zbl 1270.11070 [7] Schmidt, W. M., Diophantine approximation, 785, (1980), Springer, Berlin · Zbl 0421.10019 [8] Waldschmidt, M., Nombres transcendants, (1974), Springer-Verlag, Berlin · Zbl 0302.10030 [9] Waldschmidt, M., Diophantine approximation on linear algebraic groups : Transcendence properties of the exponential function in several variables, 326, (2000), Springer, Berlin · Zbl 0944.11024
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