# Effective Lindemann–Weierstrass theorem

The Lindemann–Weierstrass theorem states that if $\alpha_1, ..., \alpha_n$ are algebraic numbers which are linearly independent over the rational numbers $ℚ$, then $e^{\alpha_1}, ..., e^{\alpha_n}$ are algebraically independent over ℚ.

An equivalent formulation by Baker is the following: If $\alpha_1, ..., \alpha_n$ are distinct algebraic numbers, then the exponentials $e^{\alpha_1}, ..., e^{\alpha_n}$are linearly independent over the algebraic numbers. This means that $\sum_i \beta_i e^{\alpha_i} \neq 0$.

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.

This seems to be addressed in the paper by Sert (available for free on the interwebs, it seems).

Alain Sert, MR 1688184 Une version effective du théorème de Lindemann-Weierstrass par les déterminants d’interpolation, J. Number Theory 76 (1999), no. 1, 94--119.

• It is in French, but it looks to me that the result in the paper requires that $\alpha_i$'s are linear independent in Q, but here I only assume they are distinct. Hence the result is more akin to the first formulation of the L-W, but my question is akin to the Baker's formulation. I guess one can "translate" the result, but I do not know how (partially because I cannot really read French). Aug 2, 2016 at 13:57

Actually Baker's theorem generalizes Lindemann–Weierstrass, so that alredy gives you an effective bound

$$\bigg|\sum_i \beta_i e^{\alpha_i}\bigg| > Ce^{-(\log H)^k}$$

with $C$ an effectively computable constant, and everything else as in Baker's first paper on linear forms of logarithms.

• I am very curious how this can be derived from Baker's result. Aug 2, 2016 at 13:53
• BTW, it is stated that L-W is a special case of Baker's theorem, but I do not really see this ... Aug 2, 2016 at 19:44
• Do you have any idea of how C depends on the alpha_i? And what are H and k? Mar 18, 2017 at 6:08