An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II (Q2759288)

Let \(\mathbb{K}\) be an algebraic number field of degree \(D\) over \(\mathbb{Q}\) and \(\alpha_{1},\ldots, \alpha_{n}\) non zero elements of \(\mathbb{K}\). For \(1\leq j\leq n\) let \(\log\alpha_{j}\) be a non zero value of the logarithms of \(\alpha_{j}\). Further let \(b_{1},\ldots,b_{n}\) be rational integers such that the number \(\Lambda=b_{1}\log \alpha_{1}+\cdots+b_{n}\log \alpha_{n}\) is not zero. Define NEWLINE\[NEWLINEA_{j}=\max\{Dh(\alpha_{j}), |\log\alpha_{j}|, 0.16\} \quad (1\leq j\leq n), NEWLINE\]NEWLINE NEWLINE\[NEWLINE B=\max\{1,\max_{1\leq j\leq n} |b_{j}|A_{j}/A_{n}\} \quad\text{and} \quad C(n)=2^{6n+20}. NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE \log|\Lambda|\geq -C(n) D^{2}A_{1}\cdots A_{n}(1+\log D)(1+\log B). NEWLINE\]NEWLINE Other similar results are also proved. NEWLINENEWLINENEWLINEIn the previous paper [Izv. Math. 62, No. 4, 723-772 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 4, 81-136 (1998; Zbl 0923.11107)] the author proved a similar estimate under an extra assumption, called the strong independence condition, namely when \(\mathbb{K}(\sqrt{\alpha}_{1},\ldots, \sqrt{\alpha}_{n}):\mathbb{K}]=2^{n}\). In earlier works on this subject, the constant \(C(n)\) had an extra \(n^{n}\) under the strong independence condition, and an extra \(n^{2n}\) in the general case. To remove these superfluous terms was a challenge for many years. In the present paper, the author introduces a clever idea which enables him to get rid of the remaining \(n^{n}\) in the general case. If the strong independence condition is not satisfied, he uses this information in the transcendence argument together with tools from the geometry of numbers to achieve in fact a sharper estimate. NEWLINENEWLINENEWLINEThe main new ideas of the paper under review and the previous one by the author are clearly explained by \textit{Yu. V. Nesterenko} in [Linear forms in logarithms of rational numbers, Cetraro Proceedings, to appear in Lecture Notes Math., Springer-Verlag]. Further more recent work on this topic is due to \textit{Alexencev} [Linear forms in logarithms of algebraic numbers, in preparation].