Algebraic relations among solutions of linear differential equations

Abstract:

Using power series methods, Harris and Sibuya [3,4] recently showed that if $k$ is an ordinary differential field of characteristic zero and $y \ne 0$ is an element of a differential extension of $k$ such that $y$ and $1/y$ satisfy linear differential equations with coefficients in $k$, then $y\prime /y$ is algebraic over $k$. Using differential galois theory, we generalize this and characterize those polynomial relations among solutions of linear differential equations that force these solutions to have algebraic logarithmic derivatives. We also show that if $f$ is an algebraic function of genus $\geq 1$ and if $y$ and $f(y)$ or $y$ and ${e^{\int y}}$ satisfy linear differential equations, then $y$ is an algebraic function.