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, 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 Zbl

@article{JTNB_1997__9_1_97_0,
     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},
     year = {1997},
     zbl = {0905.11032},
     mrnumber = {1469664},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_97_0/}
}

TY  - JOUR
AU  - Curtis D. Bennett
AU  - Josef Blass
AU  - A. M. W. Glass
AU  - David B. Meronk
AU  - Ray P. Steiner
TI  - Linear forms in the logarithms of three positive rational numbers
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1997
SP  - 97
EP  - 136
VL  - 9
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_97_0/
LA  - en
ID  - JTNB_1997__9_1_97_0
ER  -

%0 Journal Article
%A Curtis D. Bennett
%A Josef Blass
%A A. M. W. Glass
%A David B. Meronk
%A Ray P. Steiner
%T Linear forms in the logarithms of three positive rational numbers
%J Journal de théorie des nombres de Bordeaux
%D 1997
%P 97-136
%V 9
%N 1
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_97_0/
%G en
%F JTNB_1997__9_1_97_0

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. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_97_0/

Bibliographie
Cité par

[1] A. Baker, The theory of linear forms in logarithms, in Transcendence Theory: Advances and Applications" Academic Press, London (1977), 1-27. | MR | Zbl

[2] A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. reine angew. Math. 442 (1993),19-62. | MR | Zbl

[3] J. Blass, A.M.W. Glass, D.K. Manski, D.B. Meronk, and R.P. Steiner, Constants for lower bounds for linear forms in the logarithms of algebraic numbers I, II, Acta Arith. 55 (1990), 1-22, corrigendum, ibid 65 (1993). | MR | Zbl

[4] D.W. Brownawell and D.W. Masser, Multiplicity estimates for analytic functions II, Duke Math. Journal 47 (1980), 273-295. | MR | Zbl

[5] L. Denis, Lemmes des zéros et intersections., Approximations diophantiennes et nombres transcendants Luminy (1990), éd. P. Philippon, de Gruyter (1992), 99-104. | MR | Zbl

[6] Dong Ping Ping, Minorations de combinaisons linéaires de logarithmes de nombres algébriques p-adiques, C. R. Acad. Sci. Paris, Sér.1315 (1992), 103-106. | Zbl

[7] A.O. Gel'Fond, Transcendental and algebraic numbers, (Russian). English trans.: Dover, New York (1960). | MR | Zbl

[8] A.M.W. Glass, D.B. Meronk, T. Okada, and R. Steiner, A small contribution to Catalan's equation, J. Number Theory 47 (1994), 131-137. | MR | Zbl

[9] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer Verlag, Heidelberg, 1977. | MR | Zbl

[10] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry (German), English trans.: Birkhaüser, Boston (1985). | MR | Zbl

[11] M. Laurent, Sur quelques résultats récents de transcendance, Astérisque 198-200 (1991), 209-230. | MR | Zbl

[12] M. Laurent, Hauteurs de matrices d'interpolation, Approximations diophantiennes et nombres transcendants, Luminy (1990), éd. P. Philippon, de Gruyter (1992), 215-238. | MR | Zbl

[13] M. Laurent, Linear forms in two logarithms and interpolation determinants, Acta Arith. LXVI (1994), 181-199, or Appendix to [33]. | MR | Zbl

[14] M. Laurent, M. Mignotte, and Y.V. Nesterenko, Formes linéaires en deux logarithmes et determinants d'interpolation, J. Number Theory 55 (1995), 285-321. | MR | Zbl

[15] D.W. Masser, On polynomials and exponential polynomials in several variables, Inv. Math. 63 (1981), 81-95. | MR | Zbl

[16] D.W. Masser and G. Wüstholz, Zero estimates on group varieties I, Inv. Math. 64 (1981), 489-516. | MR | Zbl

[17] D.W. Masser and G. Wüstholz, Zero estimates on group varieties II, Inv. Math. 80 (1985), 233-267. | MR | Zbl

[18] D.W. Masser and G. Wüstholz, Fields of large transcendence degree, Inv. Math. 72 (1983), 407-464. | MR | Zbl

[19] M. Mignotte and M. Waldschmidt, Linear forms in two logarithms and Schneider's method III, Ann. Fac. Sci. Toulouse 97 (1989), 43-75. | Numdam | MR | Zbl

[20] Y.V. Nesterenko, Estimates for the orders of zeros of functions of a certain class and their applications in the theory of transcendental numbers, Math. USSR Izv. 11 (1977), 239-270. | Zbl

[21] P. Philippon, Lemmes de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France 114 (1986), 355-383, et 115 (1987), 397-398. | Numdam | MR | Zbl

[22] P. Philippon and M. Waldschmidt, Lower bounds for linear forms in logarithms, Chapter 18 of New Advances in Transcendence Theory, Proc. Conf. Durham (1986), ed. A. Baker, Cambridge Univ. Press, Cambridge (1988), 280-312. | MR | Zbl

[23] P. Philippon and M. Waldschmidt, Formes linéaires de logarithmes elliptiques et mesures de transcendance, Théorie des nombres, Proc. Conf. Québec City 1987, de Gruyter, Berlin, (1989), 798-805. | MR | Zbl

[24] A.J. Van Der Poorten, Linear forms in logarithms in the p-adic case, Transcendence Theory: Advances and Applications, Academic Press, London (1977), 29-57. | MR | Zbl

[25] E. Reyssat, Approximation algébrique de nombres liés aux fonctions elliptiques et exponentielle, Bull. Soc. Math. France 108 (1980), 47-79. | Numdam | MR | Zbl

[26] T. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Tracts in Mathematics, No. 87, Cambridge Univ. Press, Cambridge (1986). | MR | Zbl

[27] D. Sinnou, Minorations de formes linéaires de logarithmes elliptiques., Publ. Math. de l'Univ. Pierre et Marie Curie, No. 106, Problèmes diophantiennes 1991-1992, exposé 3.

[28] N. Tzanakis and B.M.M. De Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), 99-132. | MR | Zbl

[29] M. Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257-283. | MR | Zbl

[30] M. Waldschmidt, Nouvelles méthodes pour minorer des combinaisons linéaires de logarithmes de nombres algébriques, Sém. Th. Nombres Bordeaux 3 (1991), 129-185. | Numdam | MR | Zbl

[31] M. Waldschmidt, Nouvelles méthodes pour minorer des combinaisons linéaires de logarithmes de nombres algébriques II, Problèmes Diophantiens 1989-1990, Publ. Univ. Pierre et Marie Curie (Paris VI) 93 (1991), 1-36.

[32] M. Waldschmidt, Minorations de combinaisons linéaires de logarithmes de nombres algébriques, Canadian J. Math 45 (1993), 176-224. | MR | Zbl

[33] M. Waldschmidt, Linear independence of logarithms of algebraic numbers, Matscience Lecture Notes, Madras (1992). | Zbl

[34] G. Wüstholz, Recent progress in transcendence theory, Springer Lecture Notes in Math., Springer Verlag, Heidelberg, 1068 (1984), 280-296. | MR | Zbl

[35] G. Wüstholz, A new approach to Baker's theorem on linear forms in logarithms I, Lecture Notes in Math., Springer Verlag, Heidelberg 1290 (1987), 189-202. | MR | Zbl

[36] G. Wüstholz, A new approach to Baker's theorem on linear forms in logarithms II, Lecture Notes in Math., Springer Verlag, Heidelberg 1290 (1987), 203-211. | MR | Zbl

[37] G. Wüstholz, A new approach to Baker's theorem on linear forms in logarithms, III, Chapter 25 of New Advances in Transcendence Theory, Proc. Conf. Durham (1986), ed. A. Baker, Cambridge Univ. Press, Cambridge (1988), 399-410. | MR | Zbl

[38] Kunrui Yu., Linear forms in p-adic logarithms I., Acta Arith. 53 (1989), 107-186. | MR | Zbl

[39] Kunrui Yu, Linear forms in p-adic logarithms II, Compositio Math. 74 (1990), 15-113. | Numdam | MR | Zbl