Non-holonomic Ideals in the Plane and Absolute Factoring

Details Non-holonomic Ideals in the Plane and Absolute Factoring Non-holonomic Ideals in the Plane and Absolute Factoring

2008-11-09

arxiv.org/abs/0811.1368v1

Mathematics

Analysis of PDEs

Scientific paper

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 $

$ 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 $

$ to have a finite number of maximal non-holonomic overideals.

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