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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.

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