Linear forms in the logarithms of three positive rational numbers
Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 97-136.

In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $\Lambda ={b}_{2}log{\alpha }_{2}-{b}_{1}log{\alpha }_{1}-{b}_{3}log{\alpha }_{3}\ne 0$ with ${b}_{1},{b}_{2},{b}_{3}$ positive integers and ${\alpha }_{1},{\alpha }_{2},{\alpha }_{3}$ positive multiplicatively independent rational numbers greater than $1$. Let ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$ with ${\alpha }_{j1},{\alpha }_{j2}$ coprime positive integers $\left(j=1,2,3\right)$. Let ${\alpha }_{j}\ge \phantom{\rule{4pt}{0ex}}\text{max}\left\{{\alpha }_{j1},e\right\}$ and assume that gcd$\left({b}_{1},{b}_{2},{b}_{3}\right)=1.$ Let

 ${b}^{\text{'}}=\left(\frac{{b}_{2}}{log{\alpha }_{1}}+\frac{{b}_{1}}{log{\alpha }_{2}}\right)\phantom{\rule{0.166667em}{0ex}}\left(\frac{{b}_{2}}{log{\alpha }_{3}}+\frac{{b}_{3}}{log{\alpha }_{2}}\right)$
and assume that $B\ge \phantom{\rule{4pt}{0ex}}\text{max}\left\{10,log{b}^{\text{'}}\right\}.$ We prove that either $\left\{{b}_{1},{b}_{2},{b}_{3}\right\}$ is $\left({c}_{4},B\right)$-linearly dependent over $ℤ$ (with respect to $\left({a}_{1},{a}_{2},{a}_{3}\right)$) or
 $\Lambda >exp\left\{-C{B}^{2}\left(\prod _{j=1}^{3}log{a}_{j}\right)\right\},$
where ${c}_{4}$ and $C={c}_{1}{c}_{2}log\rho +\delta$ are given in the tables of Section 6. Here ${b}_{1},{b}_{2},{b}_{3}$ are said to be $\left(c,B\right)$-linearly dependent over $ℤ$ if ${d}_{1}{b}_{1}+{d}_{2}{b}_{2}+{d}_{3}{b}_{3}=0$ for some ${d}_{1},{d}_{2},{d}_{3}\in ℤ$ not all $0$ with either (i) $0<|{d}_{2}|\le cBlog{a}_{2}\phantom{\rule{4pt}{0ex}}\text{min}\left\{log{a}_{1},log{a}_{3}\right\},|{d}_{1}|,|{d}_{3}|\le cBlog{a}_{1},log{a}_{3},$ or (ii) ${d}_{2}=0\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}|{d}_{1}|\le cBlog{a}_{1}log{a}_{2}\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}|{d}_{3}|\le cBlog{a}_{2}log{a}_{3}.$ In particular, we obtain ${c}_{4}<9146$ and $C<422,321$ for all values of $B\ge 10$, and for $B\ge 100$ we have ${c}_{4}\le 5572$ and $C\le 260,690.$. More complete information is given in the tables in Section 6. We prove this theorem by modifying the methods of P. Philippon, M. Waldschmidt, G. Wüstholz, et al. In particular, using a combinatorial argument, we prove that either a certain algebraic variety has dimension $0$ or $\left\{{b}_{1},{b}_{2},{b}_{3}\right\}$ are linearly dependent over $ℤ$ where the dependence has small coefficients. This allows us to improve Philippon’s zero estimate, leading to the interpolation determinant being non-zero under weaker conditions.

Dans cet article, nous donnons une minoration de la dépendance linéaire de trois nombres rationnels positifs valable sous certaines conditions faibles d’indépendance linéaire des coefficients des formes linéaires. Soit $\Lambda ={b}_{2}log{\alpha }_{2}-{b}_{1}log{\alpha }_{1}-{b}_{3}log{\alpha }_{3}\ne 0$ avec ${b}_{1},{b}_{2},{b}_{3}$ des entiers positifs et ${\alpha }_{1},{\alpha }_{2},{\alpha }_{3}$ des rationnels multiplicativement indépendants supérieurs à $1$. Soit ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$${\alpha }_{j1},{\alpha }_{j2}$ sont des entiers positifs premiers entre eux $\left(j=1,2,3\right).$ Soit ${\alpha }_{j}\ge \phantom{\rule{4pt}{0ex}}\text{max}\left\{{\alpha }_{j1},e\right\}$ et supposons que pgcd$\left({b}_{1},{b}_{2},{b}_{3}\right)=1.$ Soit

 ${b}^{\text{'}}=\left(\frac{{b}_{2}}{log{\alpha }_{1}}+\frac{{b}_{1}}{log{\alpha }_{2}}\right)\phantom{\rule{0.166667em}{0ex}}\left(\frac{{b}_{2}}{log{\alpha }_{3}}+\frac{{b}_{3}}{log{\alpha }_{2}}\right)$
et supposons que $B\ge \phantom{\rule{4pt}{0ex}}\text{max}\left\{10,log{b}^{\text{'}}\right\}.$ Nous démontrons que, soit $\left\{{b}_{1},{b}_{2},{b}_{3}\right\}$ est $\left({c}_{4},B\right)$-linéairement dépendant sur $ℤ$ (relativement à $\left({a}_{1},{a}_{2},{a}_{3}\right)$), ou bien
 $\Lambda >exp\left\{-C{B}^{2}\left(\prod _{j=1}^{3}log{a}_{j}\right)\right\}$
${c}_{4}$ et $C={c}_{1}{c}_{2}log\rho +\delta$ sont donnés dans les tables de la Section 6. Ici nous dirons que ${b}_{1},{b}_{2},{b}_{3}$ sont $\left(c,B\right)$-lineairement dépendants sur $ℤ$ si ${d}_{1}{b}_{1}+{d}_{2}{b}_{2}+{d}_{3}{b}_{3}=0$ pour certains ${d}_{1},{d}_{2},{d}_{3}\in ℤ$ non tous nuls tels que ou bien (i) $0<|{d}_{2}|\le cBlog{a}_{2}\phantom{\rule{4pt}{0ex}}\text{min}\left\{log{a}_{1},log{a}_{3}\right\},|{d}_{1}|,|{d}_{3}|\le cBlog{a}_{1},log{a}_{3},$ ou bien (ii) ${d}_{2}=0\phantom{\rule{4pt}{0ex}}\text{et}\phantom{\rule{4pt}{0ex}}|{d}_{1}|\le cBlog{a}_{1}log{a}_{2}\phantom{\rule{4pt}{0ex}}\text{et}\phantom{\rule{4pt}{0ex}}|{d}_{3}|\le cBlog{a}_{2}log{a}_{3}.$ Nous obtenons en particulier ${c}_{4}<9146$ and $C<422321$ pour tout $B\ge 10$, et si $B\ge 100$ nous avons ${c}_{4}\le 5572$ et $C\le 260690.$ Des informations plus précises sont données dans les tables de la Section 6. Nous démontrons ce résultat en modifiant les méthodes de P. Philippon, M. Waldschmidt, G. Wüstholz, et al. En particulier, par un argument combinatoire, nous prouvons que soit une certaine variété algébrique est de dimension nulle, ou bien $\left\{{b}_{1},{b}_{2},{b}_{3}\right\}$ sont linéairement dépendants sur $ℤ$, avec de petits coefficients de dépendance. Cela nous permet d’améliorer le Lemme de zéros de Philippon, nous conduisant au fait que le déterminant d’interpolation reste non nul sous des conditions plus faibles.

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title = {Linear forms in the logarithms of three positive rational numbers},
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Curtis D. Bennett; Josef Blass; A. M. W. Glass; David B. Meronk; Ray P. Steiner. Linear forms in the logarithms of three positive rational numbers. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 97-136. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_97_0/

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