Absolute factoring of non-holonomic ideals in the plane

Abstract

We study non-holonomic overideals of a left differential ideal J ⊂ F [∂x, ∂y] in two variables where F is a differentially closed field of characteristic zero. One can treat the problem of finding non-holonomic overideals as a generalization of the problem of factoring a linear partial differential operator. The main result states that a principal ideal J = P generated by an operator P with a separable symbol symb(P ) has a finite number of maximal non-holonomic overideals; the symbol is an algebraic polynomial in two variables. 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 sufficient conditions on P in order to have a finite number of maximal non-holonomic overideals. In the Appendix we study the problem of finding non-holonomic overideals of a principal ideal generated by a second order operator, the latter being equivalent to the Laplace problem. The possible application of some of these results for concrete factorization problems is pointed out.