Transcendence Measure of \(e^{1/n}\)

For a given transcendental number \(\xi \) and for any polynomial \(P(X)=: \lambda _0+\cdots +\lambda _k X^k \in \mathbb {Z}[X]\) , we know that \( P(\xi ) \neq 0.\) Let \(k \geq 1\) and \(\omega (\xi ,k, H)\) be the infimum of the numbers \(r > 0\) satisfying the estimate \(\displaystyle \left |\lambda _0+\lambda _1 \xi +\lambda _2 \xi ^{2}+ \ldots +\lambda _k\xi ^{k}\right | > \frac {1}{H^r}, \) for all \((\lambda _0, \ldots ,\lambda _k)^T \in \mathbb {Z}^{k+1}\setminus \{\overline {0}\}\) with \(\max _{1\le i\le k} \{|\lambda _i|\} \le H\) . Any function greater than or equal to \(\omega (\xi ,k,H)\) is a transcendence measure of \(\xi \) . In this article, we find out a transcendence measure of \( e^{1/n}\) which improves the earlier results.