Algebraic independence criteria. (Critères pour l’indépendance algébrique.) (French) Zbl 0615.10044

Philippon, Patrice

Publ. Math., Inst. Hautes Étud. Sci. 64, 5-52 (1986).

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 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 [G. V. Chudnovsky, Contributions to the theory of transcendental numbers (Math. Surv. Monogr.. 19) (1984; Zbl 0594.10024); E. Reyssat, J. Reine Angew. Math. 329, 66-81 (1981; Zbl 0459.10023); the reviewer and 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 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 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 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 J. W. S. Cassel’s counterexample, theorem XIV in [An introduction to diophantine approximation (Cambridge Univ. Press, 1957; Zbl 0077.04801)].

Reviewer: M.Waldschmidt

MSC:

11J85

Algebraic independence; Gel’fond’s method

Keywords:

algebraic independence criterion; transcendence; transcendence degree; diophantine approximation; exponential function; commutative algebraic groups

Citations:

Zbl 0048.03303; Zbl 0090.26103; Zbl 0594.10024; Zbl 0459.10023; Zbl 0531.10037; Zbl 0592.10028; Zbl 0354.10026; Zbl 0567.10034; Zbl 0524.10027; Zbl 0077.04801

References:

[1]	Amice, Y.,Les nombres p-adiques, Paris,puf, coll. “sup {” , 1975.} · Zbl 0313.12104
[2]	Brownawell, W. D., On the development of Gel’fond’s method, inProc. Number Theory Carbondale 1979, Springer Lecture Notes in Math.,751 (1979), 16–44.
[3]	Brownawell, W. D., etMasser, D. W., Multiplicity estimates for analytic functions II,Duke Mathematical Journal,47 (1980), 273–295. · Zbl 0461.10027 · doi:10.1215/S0012-7094-80-04718-3
[4]	Cassels, J. W. S.,An introduction to diophantine approximation, Cambridge University Press, 1957. · Zbl 0077.04801
[5]	Chudnovsky, G. V., Criteria of algebraic independence of several numbers, dansThe Riemann problem, complete integrability and arithmetic applications, Proc. 1979–1980, ed. D. V. et G. V. Chudnovsky,Springer Lecture Notes in Math.,925 (1982), 323–368. · doi:10.1007/BFb0093517
[6]	Dvornicich, R., A criterion for the algebraic dependence of two complex numbers,Bolletino UMI,15-A (1978), 678–687. · Zbl 0393.10039
[7]	Gel’fond, A. O.,Transcendental and algebraic numbers, Moscou,Gittl, 1952, et New York, Dover, 1960.
[8]	Macaulay, F. S.,The algebraic theory of modular systems, New York et Londres, Stechert-Hafner Service Agency, 1964.
[9]	Nesterenko, Y. V., Estimates for the orders of zeros of functions of a certain class and their applications in the theory of transcendental numbers,Izv. Akad. Nauk. USSR ser. Math.,41 (1977);Math. USSR Izv.,11 (1977), 239–270. · Zbl 0378.10022 · doi:10.1070/IM1977v011n02ABEH001710
[10]	Nesterenko, Y. V., On a sufficient criterion of algebraic independence of numbers,Vestnik Univ. Moscou Ser. 1, Math. Mec.,12, no 4 (1983), 63–68 (en russe). · Zbl 0524.10027
[11]	Nesterenko, Y. V., On the algebraic independence of algebraic numbers to algebraic powers, dansApproximations diophantiennes et nombres transcendants, Ed.D. Bertrand etM. Waldschmidt, C. R. Conf. Luminy, 1982,Birkhaüser Progress in Math.,31 (1983), 199–220.
[12]	Northcott, D. G.,Lessons on rings, modules and multiplicities, Cambridge University Press, 1968. · Zbl 0159.33001
[13]	Philippon, P., Indépendance algébrique de valeurs de fonctions exponentiellesp-adiques,J. für die Reine und Angew. Math.,329 (1981), 42–51. · Zbl 0459.10024 · doi:10.1515/crll.1981.329.42
[14]	Philippon, P.,Pour une théorie de l’indépendance algébrique, Thèse d’État de l’Université de Paris XI, juin 1983.
[15]	Reyssat, E., Un critère d’indépendance algébrique,J. für die Reine und Angew. Math.,329 (1981), 66–81. · Zbl 0459.10023 · doi:10.1515/crll.1981.329.66
[16]	Rudin, W.,Function theory in polydiscs, W. A. Benjamin Inc., 1969. · Zbl 0177.34101
[17]	Schneider, T.,Einfürhrung in die Transzendenten Zahlen, Grundlehren der Math. Wiss. 81, Springer Verlag, 1957; Gauthier-Villars, 1959. · Zbl 0077.04703
[18]	Waldschmidt, M., Groupes algébriques et grands degrés de transcendance,Acta Mathematica,156 (1986), 253–302. · Zbl 0592.10028 · doi:10.1007/BF02399205
[19]	Waldschmidt, M. etZhu Yao Chen, Une généralisation en plusieurs variables d’un critère de transcendance de Gel’fond,C. R. Acad. Sc. Paris Ser. I,297 (1983), 229–232.