Non-holonomic Ideals in the Plane and Absolute Factoring

TL;DR

This paper investigates the structure of non-holonomic overideals in differential ideals in two variables, establishing conditions for finiteness and infiniteness of maximal overideals based on the properties of the generating operator.

Contribution

It provides a new characterization of when principal and non-holonomic ideals have finitely many maximal overideals, especially relating to the separability of the symbol and order of the operator.

Findings

Principal ideals with separable symbol have finitely many maximal non-holonomic overideals.

Second-order operators generate ideals with infinitely many overideals iff they are essentially ordinary.

Sufficient conditions are given for third-order operators to have finitely many maximal overideals.

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.