Algebraic independence criteria.
A special case of the main theorem of this paper is the following. Let n be a positive integer. There exists \(c_ 0=c_ 0(n)>0\) with the following property. Let \(\eta >0\), and \(\theta =(\theta_ 1,...,\theta_ n)\in {\mathbb{C}}^ n\); assume that there exists a sequence \(I_ N=(P_{1,N},...,P_{m(N),N})\), \(N\geq N_ 0\) of ideals of \({\mathbb{Z}}[X_ 1,...,X_ n]\), whose set of zeros in the ball \(B(\theta, \exp (-3C_ 0N^{\eta}))\) of \({\mathbb{C}}^ n\) is finite, and such that, for each \(N\geq N_ 0\), \[ t(P_{i,N})\leq N\quad (1\leq i\leq m(N))\quad and\quad 0<\max \{| P_{i,N}(\theta)|; 1\leq i\leq m(N)\}\leq \exp (-C_ 0N^{\eta}). \] Then \(\eta <n+1.\) The case \(n=1\) is due to \textit{A. O. Gel'fond} [Transcendental and algebraic numbers (Moscou GITTL 1952; Zbl 0048.03303, and Dover-New York 1960; Zbl 0090.26103)]. In the case \(n\geq 1\), the conclusion \(\eta <2^ n\) was already known, thanks to Chudnovsky's semi-resultant [\textit{G. V. Chudnovsky}, Contributions to the theory of transcendental numbers (Math. Surv. Monogr.. 19) (1984; Zbl 0594.10024); \textit{E. Reyssat}, J. Reine Angew. Math. 329, 66-81 (1981; Zbl 0459.10023); the reviewer and \textit{Zhu Yaochen}, C. R. Acad. Sci., Paris, Sér. I 297, 229-232 (1983; Zbl 0531.10037)]. The conclusion \(\eta <n+1\) had already been derived by the author under the stronger hypothesis that the set of zeros of the homogeneous ideal associated with \(I_ N\) is of zero dimension in all of \({\mathbb{P}}_ n({\mathbb{C}})\) [the author, Pour une théorie de l'indépendance algébrique, Thèse d'Etat, Paris XI, 1983]. The new criterion has very interesting consequences. For instance the author quotes the following: if \(\alpha\) and \(\beta\) are algebraic numbers, with \(\alpha\neq 0\), log \(\alpha\neq 0\) and \(\beta\) of degree \(d\geq 2\) over \({\mathbb{Q}}\), then at least [d/2] of the numbers \(\alpha^{\beta},\alpha^{\beta^ 2},...,\alpha^{\beta^{d-1}}\) are algebraically independent. It should be noted that \textit{G. Diaz} recently [Grands degrés de transcendance pour des familles d'exponentielles, J. Number Theory (to appear)] refined this result: he replaces d/2 by \((d+1)/2.\) As shown by the author, further results of algebraic independence for values of the exponential function can be derived from his criterion. This criterion is also one of the main tools which enable one to derive results of algebraic independence for numbers related to n-parameters subgroups of commutative algebraic groups [the reviewer, Acta Math. 156, 253-302 (1986; Zbl 0592.10028)]. The proof of the criterion rests upon developments of elimination technics introduced by \textit{Yu. V. Nesterenko} [Izv. Akad. Nauk. SSSR, Ser. Mat. 41, 253-284 (1977; Zbl 0354.10026); translated in Math. USSR, Izv. 11, 239-270 (1977)]. The criterion has been sharpened by the author in order to give measures for algebraic independence [Théorie des Nombres, Delange-Pisot.-Poitou, Sémin., Paris 1983-84, Prog. Math. 59, 219-233 (1985; Zbl 0567.10034); see also \textit{Yu. V. Nesterenko}, Vestn. Mosk. Univ., Ser. I 1983, No.4, 63-68 (1983; Zbl 0524.10027)]. In an appendix, the author shows that his result is sharp: he constructs special sequences of polynomials in the spirit of \textit{J. W. S. Cassel}'s counterexample, theorem XIV in [An introduction to diophantine approximation (Cambridge Univ. Press, 1957; Zbl 0077.04801)].
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