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**On algebraic independence of algebraic powers of algebraic numbers.**
*(Russian)*
Zbl 0549.10023

This paper contains a complete proof of the following theorem. Let \(\alpha\neq 0,1\) be algebraic, let \(\beta\) be algebraic of degree \(d\geq 2\), and let t be the transcendence degree over \({\mathbb{Q}}\) of the field generated by the numbers (*) \(\alpha^{\beta},...,\alpha^{\beta^{d- 1}}\). Then \(t\geq\log_ 2(d+1);\) in other words, among the numbers (*) there are at least \([\log_ 2(d+1)]\) which are algebraically independent.

The fact that \(t\geq 1\) follows from the classical Gel’fond-Schneider Theorem (actually this gives the transcendence of every number in (*)), and the fact that \(t\geq 2\) for \(d\geq 3\) is a well-known result of Gel’fond (however, it is still not known if \(2^{2^{1/4}}\), \(2^{2^{1/2}}\) are algebraically independent). The result \(t\geq 3\) for \(d\geq 7\) was proved by G. V. Chudnovsky [Mat. Zametki 15, 661-672 (1974; Zbl 0295.10027)], and Chudnovsky’s proof that \(t\geq 4\) for \(d\geq 23\) was published by M. Waldschmidt [Sémin. Delange-Pisot-Poitou, 17e Année, 1975/76, No.G 21 (1977; Zbl 0348.10024)]. Intermediate results were also obtained by A. A. Shmelev [Mat. Zametki 11, 635- 644 (1972; Zbl 0254.10029)], W. D. Brownawell [Trans. Am. Math. Soc. 210, 1-26 (1975; Zbl 0312.10022)], and M. Waldschmidt [Sémin. Delange-Pisot-Poitou, 16e Année, 1974/75, No.G 8 (1975; Zbl 0324.10031)].

Actually in 1974 Chudnovsky announced some general results which imply the author’s theorem above, but no-one has yet succeeded in independently verifying his proofs. Following Chudnovsky’s method, P. Philippon [J. Reine Angew. Math. 329, 42-51 (1981; Zbl 0459.10024)] proved that \(t\geq\log_ 2(d+1)-1\) in the p-adic case. He used a criterion of algebraic independence due to E. Reyssat [ibid. 329, 66-81 (1981; Zbl 0459.10023)].

The author’s proof is based on different ideas taken from his fundamental paper [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253-284 (1977; Zbl 0354.10026); translated as Math. USSR, Izv. 11, 239-270 (1977)] together with his explicit estimates for Hilbert functions [Mat. Sb., Nov. Ser. 123 (165), 11-34 (1983)]. By means of his U-elimination method he defines the notions of the ”taille” of a polynomial ideal analogous to that of an ordinary polynomial (i.e., the sum of the degree and the logarithmic height), and also the value of a polynomial ideal at a point. These are shown to behave well with respect to the standard operations of commutative algebra, such as decomposition into primary ideals, etc. He then states a quantitative result estimating from below the value of an arbitrary ideal I at (*) in terms of its taille. This is proved by induction on the dimension of I. The main theorem above is an immediate consequence for suitably large dimension. It may be worth remarking that the latter parts of the proof have meanwhile appeared in English [Approximations diophantiennes et nombres transcendants, Colloq. Luminy/Fr. 1982, Prog. Math. 31, 199-220 (1983; Zbl 0513.10035)].

[Recently P. Philippon, in a paper to appear in Publ. Math., Inst. Hautes Étud. Sci., has used the author’s ideas to prove that in fact \(t\geq [d/2]\). Since trivially \(t\leq d-1\), this may be regarded as ”one- half” of the natural conjecture that \(t=d-1\) always.]

The fact that \(t\geq 1\) follows from the classical Gel’fond-Schneider Theorem (actually this gives the transcendence of every number in (*)), and the fact that \(t\geq 2\) for \(d\geq 3\) is a well-known result of Gel’fond (however, it is still not known if \(2^{2^{1/4}}\), \(2^{2^{1/2}}\) are algebraically independent). The result \(t\geq 3\) for \(d\geq 7\) was proved by G. V. Chudnovsky [Mat. Zametki 15, 661-672 (1974; Zbl 0295.10027)], and Chudnovsky’s proof that \(t\geq 4\) for \(d\geq 23\) was published by M. Waldschmidt [Sémin. Delange-Pisot-Poitou, 17e Année, 1975/76, No.G 21 (1977; Zbl 0348.10024)]. Intermediate results were also obtained by A. A. Shmelev [Mat. Zametki 11, 635- 644 (1972; Zbl 0254.10029)], W. D. Brownawell [Trans. Am. Math. Soc. 210, 1-26 (1975; Zbl 0312.10022)], and M. Waldschmidt [Sémin. Delange-Pisot-Poitou, 16e Année, 1974/75, No.G 8 (1975; Zbl 0324.10031)].

Actually in 1974 Chudnovsky announced some general results which imply the author’s theorem above, but no-one has yet succeeded in independently verifying his proofs. Following Chudnovsky’s method, P. Philippon [J. Reine Angew. Math. 329, 42-51 (1981; Zbl 0459.10024)] proved that \(t\geq\log_ 2(d+1)-1\) in the p-adic case. He used a criterion of algebraic independence due to E. Reyssat [ibid. 329, 66-81 (1981; Zbl 0459.10023)].

The author’s proof is based on different ideas taken from his fundamental paper [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 253-284 (1977; Zbl 0354.10026); translated as Math. USSR, Izv. 11, 239-270 (1977)] together with his explicit estimates for Hilbert functions [Mat. Sb., Nov. Ser. 123 (165), 11-34 (1983)]. By means of his U-elimination method he defines the notions of the ”taille” of a polynomial ideal analogous to that of an ordinary polynomial (i.e., the sum of the degree and the logarithmic height), and also the value of a polynomial ideal at a point. These are shown to behave well with respect to the standard operations of commutative algebra, such as decomposition into primary ideals, etc. He then states a quantitative result estimating from below the value of an arbitrary ideal I at (*) in terms of its taille. This is proved by induction on the dimension of I. The main theorem above is an immediate consequence for suitably large dimension. It may be worth remarking that the latter parts of the proof have meanwhile appeared in English [Approximations diophantiennes et nombres transcendants, Colloq. Luminy/Fr. 1982, Prog. Math. 31, 199-220 (1983; Zbl 0513.10035)].

[Recently P. Philippon, in a paper to appear in Publ. Math., Inst. Hautes Étud. Sci., has used the author’s ideas to prove that in fact \(t\geq [d/2]\). Since trivially \(t\leq d-1\), this may be regarded as ”one- half” of the natural conjecture that \(t=d-1\) always.]

Reviewer: D.W.Masser

### MSC:

11J85 | Algebraic independence; Gel’fond’s method |