Non-holonomic Ideals in the Plane and Absolute Factoring

Abstract

We study {\it non-holonomic} overideals of a left differential ideal $J\subset F[{\partial }_x,{\partial }_y]$ in two variables where $F$ is a differentially closed field of characteristic zero. The main result states that a principal ideal $J=<P>$ generated by an operator $P$ with a separable {\it symbol} $symb(P)$, which is a homogeneous polynomial in two variables, has a finite number of maximal non-holonomic overideals. This statement is extended to non-holonomic ideals $J$ with a separable symbol. As an application we show that in case of a second-order operator $P$ the ideal $<P>$ has an infinite number of maximal non-holonomic overideals iff $P$ is essentially ordinary. In case of a third-order operator $P$ we give few sufficient conditions on $<P>$ to have a finite number of maximal non-holonomic overideals.