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Logarithmic forms and group varieties. (English) Zbl 0788.11026
The authors derive the following improvement of Baker’s lower bound for linear forms in logarithms of algebraic numbers. Let \(\alpha_ 1,\dots,\alpha_ n\) be algebraic numbers \(\neq 0,1\), \(K=\mathbb{Q} (\alpha_ 1, \dots,\alpha_ n)\) and \(d=[K:\mathbb{Q}]\). Suppose that \(K\) is contained in \(\mathbb{C}\) and choose for every \(\alpha\in K\) a determination of \(\log\alpha\). Define the modified height of \(\alpha\) by \(h'(\alpha)= d^{-1} \max(h(\alpha),| \log\alpha|,1)\) where \(h(\alpha)\) is the relative logarithmic Weil height relative to \(K\). Further, let \(b_ 1,\dots,b_ n\) be rational integers not all zero and put \(\Lambda=b_ 1 \log\alpha_ 1 +\cdots+ b_ n\log \alpha_ n\) and \(L=d^{-1} \max(1,\log | b_ 1|,\dots, \log| b_ n|)\). The authors prove that if \(\Lambda\neq 0\) then \[ \log| \Lambda|>- C(n,d)h'(\alpha_ 1) \cdots h'(\alpha_ n)L, \] where \(C(n,d)= 18(n+1)! n^{n+1}(32d)^{n+2} \log(2nd)\). Compared with previous estimates, a factor \(\log(h'(\alpha_ 1) \cdots h'(\alpha_{n-1}))\) has been removed, and the constant \(C(n,d)\) is much smaller. The main new ingredient in the proof is G. Wüstholz’s multiplicity estimate [Ann. Math., II. Ser. 129, 471-500 (1989; Zbl 0675.10024)]; moreover, compared with previous proofs, the Kummer descent and the extrapolation technique have been improved.

11J86 Linear forms in logarithms; Baker’s method
14L10 Group varieties
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