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Mathematics of Computation

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Explicit lower bounds for linear forms

Author: Kalle Leppälä
Journal: Math. Comp. 85 (2016), 2995-3008
MSC (2010): Primary 11J25, 11J82
Published electronically: January 19, 2016
MathSciNet review: 3522978
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Abstract: Let $ \mathbb{I}$ be the field of rational numbers or an imaginary quadratic field and $ \mathbb{Z}_\mathbb{I}$ its ring of integers. We study some general lemmas that produce lower bounds

$\displaystyle \lvert B_0+B_1\theta _1+\cdots +B_r\theta _r\rvert \ge \frac {1}{\max \{\lvert B_1 \rvert ,\ldots ,\lvert B_r \rvert \}^\mu } $

for all $ B_0,\ldots ,B_r \in \mathbb{Z}_{\mathbb{I}}$, $ \max \{\lvert B_1 \rvert ,\ldots ,\lvert B_r \rvert \} \ge H_0$, given suitable simultaneous approximating sequences of the numbers $ \theta _1,\ldots ,\theta _r$. We manage to replace the lower bound with $ 1/{\max \{\lvert B_1 \rvert ^{\mu _1},\ldots ,\lvert B_r \rvert ^{\mu _r}\}}$ for all $ B_0,\ldots ,B_r \in \mathbb{Z}_{\mathbb{I}}$, $ \max \{\lvert B_1 \rvert ^{\mu _1},\ldots ,\lvert B_r \rvert ^{\mu _r}\} \ge H_0$, where the exponents $ \mu _1,\ldots ,\mu _r$ are different when the given type II approximating sequences approximate some of the numbers $ \theta _1,\ldots ,\theta _r$ better than the others. As an application we research certain linear forms in logarithms. Our results are completely explicit.

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Kalle Leppälä
Affiliation: Bioinformatics Research Centre (BIRC), University of Århus, Denmark

Received by editor(s): November 4, 2014
Received by editor(s) in revised form: April 11, 2015, and May 7, 2015
Published electronically: January 19, 2016
Article copyright: © Copyright 2016 American Mathematical Society