An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers.

*(English. Russian original)*Zbl 0923.11107
Izv. Math. 62, No. 4, 723-772 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 4, 81-136 (1998).

The main goal of this paper is to provide a sharp explicit lower bound for non vanishing numbers of the form
\[
\Lambda=b_{1}\log\alpha_{1}+\cdots+ b_{n}\log\alpha_{n},
\]
when \(b_{1},\ldots,b_{n}\) are rational integers and \(\alpha_{1},\ldots,\alpha_{n}\) are non-zero algebraic numbers. The introduction provides a very comprehensive survey of previous work on this topic (with relevant discussion on the different methods and comparisons of the results), including an estimate which was announced by the author in 1983 [E. M. Matveev, An estimate of a linear form in the logarithms of algebraic numbers; abstracts of the conference “The theory of transcendental numbers and its applications”, Moscow 2.–4.02.1983, Moscow State Univ. Press, Moscow 1983 (in Russian)]. This estimate is valid only under a so-called condition of strong independence, namely when the number field \(K=\mathbb Q (\alpha_{1},\ldots,\alpha_{n})\) satisfies \([K(\sqrt{\alpha_{1}},\ldots, \sqrt{\alpha_{n}}):K]=2^{n}\). Under this hypothesis, his estimate was very sharp, especially in terms of the parameter \(n\): the main feature is that, in place of \(n^{n}\) which was present in the previous results, a single exponential \(c^{n}\) is there, with an explicit constant \(c\).

In the present paper the author produces a proof not only of this estimate, but even of a refined one. The statement of the main result involves too heavy a notation to be quoted here (for instance not less than six parameters \(B\), \(B_{0}\), \(B_{1}\), \(B_{2}\), \(B_{3}\) and \(W\) are used to measure the height of the coefficients \(b_{1},\ldots,b_{n}\)). However the new idea in the proof (along Baker’s method) is neat. It bears some analogy with C. L. Stewart’s trick in [Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. Fr. 106, 169-176 (1978; Zbl 0396.12002)].

A \(p\)-adic analog of this paper has been worked out by K. Yu [Acta Arith. 89, No. 4, 337-378 (1999; Zbl 0928.11031)]. Such estimates are relevant for applications to diophantine results. An application related to the \(abc\) conjecture is due to C. L. Stewart and K. Yu [On the abc conjecture. II (in preparation)].

In the present paper the author produces a proof not only of this estimate, but even of a refined one. The statement of the main result involves too heavy a notation to be quoted here (for instance not less than six parameters \(B\), \(B_{0}\), \(B_{1}\), \(B_{2}\), \(B_{3}\) and \(W\) are used to measure the height of the coefficients \(b_{1},\ldots,b_{n}\)). However the new idea in the proof (along Baker’s method) is neat. It bears some analogy with C. L. Stewart’s trick in [Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. Fr. 106, 169-176 (1978; Zbl 0396.12002)].

A \(p\)-adic analog of this paper has been worked out by K. Yu [Acta Arith. 89, No. 4, 337-378 (1999; Zbl 0928.11031)]. Such estimates are relevant for applications to diophantine results. An application related to the \(abc\) conjecture is due to C. L. Stewart and K. Yu [On the abc conjecture. II (in preparation)].

Reviewer: M.Waldschmidt (Paris)