DIFFERENTIAL SIMPLICITY IN POLYNOMIAL RINGS AND ALGEBRAIC INDEPENDENCE OF POWER SERIES

Abstract

Let $k$ be a field of characteristic zero, $f(X,Y), g(X,Y)\,{\in}\,k[X,Y]$, $g(X,Y)\,{\notin}\,(X,Y)$ and $d\,{:=} g(X,Y)\,{\partial}/ {\partial X}+f(X,Y)\,{\partial}/{\partial Y}$. A connection is established between the $d$-simplicity of the local ring $k[X,Y]_{(X,Y)}$ and the transcendency of the solution in $tk[[t]]$ of the algebraic differential equation $g(t,y(t))\,{\cdot}\, ({\partial}/{\partial t})y(t)=f(t,y(t))$. This connection is used to obtain some interesting results in the theory of the formal power series and to construct new examples of differentially simple rings.