Linear forms in the logarithms of three positive rational numbers
Let \(\alpha_{1},\alpha_{2},\alpha_{3}\) be multiplicatively independent rational numbers, each \(>1\); let \(b_{1},b_{2},b_{3}\) be positive integers with \(b_{2}\log\alpha_{2}>\max\{b_{1}\log\alpha_{1},b_{3}\log\alpha_{ 3}\}\), such that the number \(\Lambda=b_{2}\log\alpha_{2}-b_{1}\log\alpha_{1}-b_{3}\log\alpha_{3}\) does not vanish. Write \(\alpha_{j}=\alpha_{j1}/\alpha_{j2}\) and \(a_{j}=\max\{\alpha_{j1},e\}\) \ (\(j=1,2\)). Further, define \[ b'=\left({b_{2}\over\log a_{1}}+{b_{1}\over\log a_{2}}\right) \left({b_{2}\over\log a_{3}}+{b_{3}\over\log a_{2}}\right) \] and \(B=\max\{10,\log b'\}\). The conclusion of Theorem 1 is then \[ | \Lambda| \geq\exp\{-4.5\cdot 10^{5} B^{2}\log a_{1}\log a_{2}\log a_{3}\}. \] There is however a further technical hypothesis, namely that \(\gcd(b_{1},b_{2},b_{3})=1\) and \(d_{1}b_{1}+d_{2}b_{2}+d_{3}b_{3}\not=0\) for any triple \((d_{1},d_{2},d_{3})\not=(0,0,0)\) of rational integers with either \[ 0<| d_{2}| \leq 10^{4}B\log a_{2}\min\{\log a_{1},\log a_{3}\} \] and \[ \max\{| d_{1}| , | d_{3}| \}\leq 10^{4}B\log a_{1}\log a_{3} \] or else \(d_{2}=0\) and \[ | d_{i}| \leq 10^{4}B\log a_{2} \log a_{i}, \quad i=1,3. \] Theorem 2 of the paper under review provides a refined estimate with better numerical values than the above \(4.5\cdot 10^{5}\) and \(10^{4}\) respectively. The proof is an extension in two variables of the work on linear forms in two logarithms by \textit{M. Laurent, M. Mignotte} and \textit{Yu. V. Nesterenko} [J. Number Theory 55, No. 2, 285-321 (1995; Zbl 0843.11036)] using Laurent's interpolation determinants [\textit{M. Laurent}, Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 209-230 (1991; Zbl 0762.11027); Acta Arith. 66, No. 2, 181-199 (1994; Zbl 0801.11034); Approximations diophantiennes et nombres transcendants, C.-R. Colloq., Luminy/Fr. 1990, 215-238 (1992; (Zbl 0773.11047)]. Their zero estimate involves a refinement of that of \textit{P. Philippon} [Bull. Soc. Math. Fr. 114, 355-383 (1986; Zbl 0617.14001); Errata et addenda, Bull. Soc. Math. Fr. 115, 397-398 (1987; Zbl 0634.14001)] for the special case of polynomials vanishing at a set of points of the form \[ \bigl\{ \bigl(r+sb_{1}/b_{2},t+sb_{3}/b_{2}, \alpha_{1}^{r}\alpha_{2}^{s}\alpha_{3}^{t}\bigr);\;0\leq r<R, \;0\leq s<S, \;0\leq t<T\bigr\}\subset\mathbb C^3. \] In two forthcoming papers, \textit{P. Voutier} [Some linear forms in logarithms, J. Ramanujan Math. Soc. (to appear) and Linear forms in three logarithms, Can. J. Math. (to appear)] deals more generally with three logarithms of algebraic numbers and succeeds to avoid the extra technical hypothesis.
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