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Good evening,

Given $\alpha_1,\dots,\alpha_n$ non-zero algebraic numbers. I want to find an effective bound $N$ such that if for all $(\beta_1,\dots,\beta_n)\in\mathbb{Z}^n\setminus \{0\}$ with $|\beta_i| \leq N,\; i=1,\dots, n,$ we have $\beta_1 \ln \alpha_1 +\dots + \beta_n \ln \alpha_n \neq 0$, then $\ln \alpha_1,\dots,\ln \alpha_n$ are linearly independent over $\mathbb{Q}$.

I thought it would be possible to use Baker’s Theorem, but I don't see how. It's probably an already known result, so if you find a reference, it's also good to me. Thank you in advance.

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It will depend on the primes that divide the $\alpha$'s as well as the regulator of the field defined by them. So you may not get a simple expression for $N$. Do you want an expression or an algorithm? – Felipe Voloch Sep 3 '13 at 16:30
I want an explicit bound $N$ depending on $n$ and the height of the $\alpha$'s (but not necessarly the optimal one). But yes, behind this my idea is to have an algorithm to prove linear independance using the PSLQ algorithm. – Combot Sep 3 '13 at 16:42

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