Linear forms in the logarithms of three positive rational numbers

Curtis D. Bennett; Josef Blass; A. M. W. Glass; David B. Meronk; Ray P. Steiner

doi:10.5802/jtnb.192

Curtis D. Bennett ; Josef Blass ; A. M. W. Glass ; David B. Meronk ; Ray P. Steiner

Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 97-136.

Résumé
Abstract

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 $Λ = b_{2} log α_{2} - b_{1} log α_{1} - b_{3} log α_{3} \neq 0$ avec $b_{1}, b_{2}, b_{3}$ des entiers positifs et $α_{1}, α_{2}, α_{3}$ des rationnels multiplicativement indépendants supérieurs à $1$ . Soit $α_{j 1} = α_{j 1} / α_{j 2}$ où $α_{j 1}, α_{j 2}$ sont des entiers positifs premiers entre eux $(j = 1, 2, 3) .$ Soit $α_{j} \geq max {α_{j 1}, e}$ et supposons que pgcd $(b_{1}, b_{2}, b_{3}) = 1 .$ Soit $b^{'} = (\frac{b_{2}}{log α_{1}} + \frac{b_{1}}{log α_{2}}) (\frac{b_{2}}{log α_{3}} + \frac{b_{3}}{log α_{2}})$ et supposons que $B \geq max {10, log b^{'}} .$ Nous démontrons que, soit ${b_{1}, b_{2}, b_{3}}$ est $(c_{4}, B)$ -linéairement dépendant sur $ℤ$ (relativement à $(a_{1}, a_{2}, a_{3})$ ), ou bien $Λ > exp \{- C B^{2} (\prod_{j = 1}^{3} log a_{j})\}$ où $c_{4}$ et $C = c_{1} c_{2} log ρ + δ$ sont donnés dans les tables de la Section 6. Ici nous dirons que $b_{1}, b_{2}, b_{3}$ sont $(c, B)$ -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) $0 < | d_{2} | \leq c B log a_{2} min {log a_{1}, log a_{3}}, | d_{1} |, | d_{3} | \leq c B log a_{1}, log a_{3},$ ou bien (ii) $d_{2} = 0 et | d_{1} | \leq c B log a_{1} log a_{2} et | d_{3} | \leq c B log a_{2} log a_{3} .$ Nous obtenons en particulier $c_{4} < 9146$ and $C < 422321$ pour tout $B \geq 10$ , et si $B \geq 100$ nous avons $c_{4} \leq 5572$ et $C \leq 260690 .$ 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 ${b_{1}, b_{2}, b_{3}}$ 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.

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 $Λ = b_{2} log α_{2} - b_{1} log α_{1} - b_{3} log α_{3} \neq 0$ with $b_{1}, b_{2}, b_{3}$ positive integers and $α_{1}, α_{2}, α_{3}$ positive multiplicatively independent rational numbers greater than $1$ . Let $α_{j 1} = α_{j 1} / α_{j 2}$ with $α_{j 1}, α_{j 2}$ coprime positive integers $(j = 1, 2, 3)$ . Let $α_{j} \geq max {α_{j 1}, e}$ and assume that gcd $(b_{1}, b_{2}, b_{3}) = 1 .$ Let $b^{'} = (\frac{b_{2}}{log α_{1}} + \frac{b_{1}}{log α_{2}}) (\frac{b_{2}}{log α_{3}} + \frac{b_{3}}{log α_{2}})$ and assume that $B \geq max {10, log b^{'}} .$ We prove that either ${b_{1}, b_{2}, b_{3}}$ is $(c_{4}, B)$ -linearly dependent over $ℤ$ (with respect to $(a_{1}, a_{2}, a_{3})$ ) or $Λ > exp \{- C B^{2} (\prod_{j = 1}^{3} log a_{j})\},$ where $c_{4}$ and $C = c_{1} c_{2} log ρ + δ$ are given in the tables of Section 6. Here $b_{1}, b_{2}, b_{3}$ are said to be $(c, B)$ -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) $0 < | d_{2} | \leq c B log a_{2} min {log a_{1}, log a_{3}}, | d_{1} |, | d_{3} | \leq c B log a_{1}, log a_{3},$ or (ii) $d_{2} = 0 and | d_{1} | \leq c B log a_{1} log a_{2} and | d_{3} | \leq c B log a_{2} log a_{3} .$ In particular, we obtain $c_{4} < 9146$ and $C < 422, 321$ for all values of $B \geq 10$ , and for $B \geq 100$ we have $c_{4} \leq 5572$ and $C \leq 260, 690 .$ . 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 $\{b_{1}, b_{2}, b_{3}\}$ 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.

MR 1469664 | Zbl 0905.11032

DOI : https://doi.org/10.5802/jtnb.192

@article{JTNB_1997__9_1_97_0,
     author = {Bennett, Curtis D. and Blass, Joseph and Glass, A. M. W. and Meronk, David B. and Steiner, Ray P.},
     title = {Linear forms in the logarithms of three positive rational numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {97--136},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {1},
     year = {1997},
     doi = {10.5802/jtnb.192},
     zbl = {0905.11032},
     mrnumber = {1469664},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.192/}
}

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, Tome 9 (1997) no. 1, pp. 97-136. doi : 10.5802/jtnb.192. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.192/

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