This question is a continuation of the question I posted on https://mathoverflow.net/questions/245651/effective-lindemann-weierstrass-theorem
A preciser formulation of the question is as follows:
Consider the splitting field $K=Q[\alpha_1, \cdots, \alpha_n]$ where $\alpha_1, \cdots, \alpha_n$ are roots of an irreducible polynomial $f(x)$. Assume further $\beta_1, \cdots, \beta_n\in K$. The Lindermann-Weierstrass theorem says that (modulo some trivial case) $\sum_i \beta_i e^{\alpha_i} \neq 0$.
My question is: suppose $\sum_i \beta_i e^{\alpha_i} \neq 0$, how to give a lower bound of $|\sum_i \beta_i e^{\alpha_i}|$. In other words, I am looking for some analogical result of Baker's theorem. Note that Baker's theorem is about the logarithm, i.e., a lower bound of $\sum_i \beta_i \log(\alpha_i)$, but here we have exponents.
Although there are two answers provided on the link, I am still confused. For the first answer, it seems the reference requires that $\alpha_i$'s are linear independent in $Q$, but here I only assume they are distinct. I guess one can "translate" the result, but I do not know exactly how.
For the second answer, I cannot see how it is derived from Baker's theorem.
Please help me out.