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

Journal de théorie des nombres de Bordeaux (1997)

  • Volume: 9, Issue: 1, page 97-136
  • ISSN: 1246-7405

Abstract

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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 ≠ 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 ≥ max { α j 1 , e } and assume that gcd ( b 1 , b 2 , b 3 ) = 1 . Let b ' = b 2 log α 1 + b 1 log α 2 b 2 log α 3 + b 3 log α 2 and assume that B ≥ 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 ∏ 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 ∈ ℤ not all 0 with either (i) 0 < | d 2 | ≤ c B log a 2 min { log a 1 , log a 3 } , | d 1 | , | d 3 | ≤ c B log a 1 , log a 3 , or (ii) d 2 = 0 and | d 1 | ≤ c B log a 1 log a 2 and | d 3 | ≤ c B log a 2 log a 3 . In particular, we obtain c 4 < 9146 and C < 422 , 321 for all values of B ≥ 10 , and for B ≥ 100 we have c 4 ≤ 5572 and C ≤ 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.

How to cite

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Bennett, Curtis D., et al. "Linear forms in the logarithms of three positive rational numbers." Journal de théorie des nombres de Bordeaux 9.1 (1997): 97-136. <http://eudml.org/doc/248016>.

@article{Bennett1997,
abstract = {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 $(j = 1, 2, 3)$. Let $\alpha _j \ge \text\{ max\} \lbrace \alpha _\{j1\}, e\rbrace $ and assume that gcd$(b_1, b_2, b_3) = 1.$ Let\begin\{equation*\}b^\{\prime \} = \left( \frac\{b\_2\}\{\log \alpha \_1\} + \frac\{b\_1\}\{\log \alpha \_2\} \right) \, \left( \frac\{b\_2\}\{\log \alpha \_3\} + \frac\{b\_3\}\{\log \alpha \_2\} \right) \end\{equation*\}and assume that $B \ge \text\{ max\} \lbrace 10, \log b^\{\prime \}\rbrace .$ We prove that either $\lbrace b_1, b_2, b_3\rbrace $ is $\left( c_4, B \right)$-linearly dependent over $\mathbb \{Z\}$ (with respect to $\left( a_1, a_2, a_3 \right)$) or\begin\{equation*\}\Lambda &gt; \exp \left\lbrace - CB^2 \left( \prod ^\{3\}\_\{j = 1\} \log a\_j \right) \right\rbrace , \end\{equation*\}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 $(c, B)$-linearly dependent over $\mathbb \{Z\}$ if $d_1b_1 + d_2b_2 + d_3b_3 = 0$ for some $d_1, d_2, d_3 \in \mathbb \{Z\}$ not all $0$ with either (i) $0 &lt; |d_2| \le cB \log a_2 \text\{ min\}\lbrace \log a_1, \log a_3 \rbrace , |d_1|, |d_3| \le cB \log a_1, \log a_3,$ or (ii) $d_2 = 0 \text\{ and \} |d_1| \le cB \log a_1 \log a_2 \text\{ and \} |d_3| \le cB \log a_2 \log a_3.$ In particular, we obtain $c_4 &lt; 9146$ and $C &lt; 422,321$ 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\lbrace b_1,b_2,b_3 \right\rbrace $ are linearly dependent over $\mathbb \{Z\}$ 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.},
author = {Bennett, Curtis D., Blass, Josef, Glass, A. M. W., Meronk, David B., Steiner, Ray P.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {interpolation determinants; linear forms in three logarithms; zero estimates},
language = {eng},
number = {1},
pages = {97-136},
publisher = {Université Bordeaux I},
title = {Linear forms in the logarithms of three positive rational numbers},
url = {http://eudml.org/doc/248016},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Bennett, Curtis D.
AU - Blass, Josef
AU - Glass, A. M. W.
AU - Meronk, David B.
AU - Steiner, Ray P.
TI - Linear forms in the logarithms of three positive rational numbers
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 1
SP - 97
EP - 136
AB - 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 $(j = 1, 2, 3)$. Let $\alpha _j \ge \text{ max} \lbrace \alpha _{j1}, e\rbrace $ and assume that gcd$(b_1, b_2, b_3) = 1.$ Let\begin{equation*}b^{\prime } = \left( \frac{b_2}{\log \alpha _1} + \frac{b_1}{\log \alpha _2} \right) \, \left( \frac{b_2}{\log \alpha _3} + \frac{b_3}{\log \alpha _2} \right) \end{equation*}and assume that $B \ge \text{ max} \lbrace 10, \log b^{\prime }\rbrace .$ We prove that either $\lbrace b_1, b_2, b_3\rbrace $ is $\left( c_4, B \right)$-linearly dependent over $\mathbb {Z}$ (with respect to $\left( a_1, a_2, a_3 \right)$) or\begin{equation*}\Lambda &gt; \exp \left\lbrace - CB^2 \left( \prod ^{3}_{j = 1} \log a_j \right) \right\rbrace , \end{equation*}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 $(c, B)$-linearly dependent over $\mathbb {Z}$ if $d_1b_1 + d_2b_2 + d_3b_3 = 0$ for some $d_1, d_2, d_3 \in \mathbb {Z}$ not all $0$ with either (i) $0 &lt; |d_2| \le cB \log a_2 \text{ min}\lbrace \log a_1, \log a_3 \rbrace , |d_1|, |d_3| \le cB \log a_1, \log a_3,$ or (ii) $d_2 = 0 \text{ and } |d_1| \le cB \log a_1 \log a_2 \text{ and } |d_3| \le cB \log a_2 \log a_3.$ In particular, we obtain $c_4 &lt; 9146$ and $C &lt; 422,321$ 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\lbrace b_1,b_2,b_3 \right\rbrace $ are linearly dependent over $\mathbb {Z}$ 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.
LA - eng
KW - interpolation determinants; linear forms in three logarithms; zero estimates
UR - http://eudml.org/doc/248016
ER -

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