Logarithmic Derivatives of Solutions to Linear Differential Equations

Given an ordinary differential field $K$ of characteristic zero, it is known that if $y$ and $1\mathrm{/}y$ satisfy linear differential equations with coefficients in $K$, then $y^{\mathrm{\prime }}\mathrm{/}y$ is algebraic over $K$. We present a new short proof of this fact using Gr"{o}bner basis techniques and give a direct method for finding a polynomial over $K$ that $y^{\mathrm{\prime }}\mathrm{/}y$ satisfies. Moreover, we provide explicit degree bounds and extend the result to fields with positive characteristic. Finally, we give an application of our method to a class of nonlinear differential equations.