The $n$th roots of solutions of linear ordinary differential equations

Abstract:

In this paper we shall prove the following theorem: Let $K$ be a differential field of characteristic zero. Let $\varphi$ and $\psi$ be elements of a differential field extension of $K$ such that (i) $\varphi \ne 0$ and $\psi \ne 0$; (ii) $\varphi$ and $\psi$ satisfy nontrivial linear differential equations with coefficients in $K$, say, $P(\varphi ) = 0$ and $Q(\psi ) = 0$; (iii) $\varphi = {\psi ^n}$ for some positive integer $n$ such that $n \geqslant {\text { ord }}P$. Then the logarithmic derivatives of $\varphi$ and $\psi$ are algebraic over $K$. (Note that $\varphi ’/\varphi = n(\psi ’/\psi )$.)