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

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

Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 97-136.

Abstract
Résumé

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.

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.

MR: 1469664 | Zbl: 0905.11032

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     author = {Curtis D. Bennett and Josef Blass and A. M. W. Glass and David B. Meronk and Ray P. Steiner},
     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},
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     mrnumber = {1469664},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_97_0/}
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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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