On approximations of solutions of the equation $P(z,\ln z)=0$ by algebraic numbers

Abstract

The paper is devoted to studying how well solutions of an equation $P\left(z,lnz\right)=0$, where $P\left(x,y\right)\in \mathbb{Z}\left[x,y\right]$, can be approximated with algebraic numbers. We prove a new bound with the help of a construction due to K. Mahler.