Geodesic
On the linear independence of numbers over number fields
Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 506-517 
E. V. Bedulev. On the linear independence of numbers over number fields. Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 506-517. http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a2/
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     title = {On the linear independence of numbers over number fields},
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     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper, the problem of a lower bound for the measure of linear independence of a given collection of numbers $\theta_1,\dots,\theta_n$ is considered under the assumption that, for a sequence of polynomials whose coefficients are algebraic integers, upper and lower estimates at the point $(\theta_1,\dots,\theta_n)$ are known. We use a method that generalizes the Nesterenko method to the case of an arbitrary algebraic number field.

[1] Chudnovskii G. V., Analiticheskie metody v diofantovykh priblizheniyakh, Preprint No. 74.8; Препринт No. 74.9, ИМ АН УССР, Киев, 1974

[2] Ressat E., “Un critère d'indépendance algébrique”, J. Reine Angew. Math., 329 (1981), 66–81 | MR

[3] Waldschmidt M., Zhu Yao Chen, “Une généralisation en plusieures variables d'un critère de transcendance de Gelfond”, C. R. Acad. Sci. Paris. Sér. I. Math., 297 (1983), 229–232 | MR | Zbl

[4] Nesterenko Yu. V., “Ob odnom dostatochnom priznake algebraicheskoi nezavisimosti chisel”, Vestn. MGU. Ser. 1. Matem., mekh., 1983, no. 4, 63–68 | MR | Zbl

[5] Phillipon P., “Critères pour l'indépendance algébrique de familles de nombres”, C. R. Math. Rep. Acad. Sci. Canada, 6 (1984), 285–290 | MR

[6] Nesterenko Yu. V., “O mere lineinoi nezavisimosti chisel”, Vestn. MGU. Ser. 1. Matem., mekh., 1985, no. 1, 46–49 | MR | Zbl

[7] Gantmakher R., Teoriya matrits, Nauka, M., 1986

[8] Leng S., Osnovy diofantovoi geometrii, Mir, M., 1986 | Zbl

[9] Kagan V. F., Osnovaniya teorii opredelitelei, M., 1921