Transcendence measure of $e^{1/n}$For a given transcendental number $ξ$ and for any polynomial $P(X)=:
λ_0+\cdots+λ_k X^k \in \mathbb{Z}[X]$, we know that $ P(ξ) \neq
0.$ Let $k \geq 1$ and $ω(k, H)$ be the infimum of the numbers $r > 0$
satisfying the estimate $$ \left|λ_0+λ_1 ξ+λ_2 ξ^{2}+
\ldots +λ_kξ^{k}\right| > \frac{1}{H^r}, $$ for all $(λ_0, \ldots
,λ_k)^T \in \mathbb{Z}^{k+1}\setminus\{\overline{0}\}$ with $\max_{1\le
i\le k} \{|λ_i|\} \le H$. Any function greater than or equal to $ω
(k, H)$ is a {\it transcendence measure of $ξ$}. In this article, we find out
a transcendence measure of $ e^{1/n}$ which improves a result proved by
Mahler(\cite{Mahler}) in 1975.
arxiv.org 
