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Absolute factoring of non-holonomic ideals in the plane

 
: Grigoriev, D.; Schwarz, F.

:

Watt, S.M. ; Association for Computing Machinery -ACM-, Special Interest Goup on Symbolic and Algebraic Manipulation -SIGSAM-:
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC 2010 : Munich, Germany, July 25 - 28, 2010
New York: ACM, 2010
ISBN: 978-1-4503-0150-3
S.93-97
International Symposium on Symbolic and Algebraic Computation (ISSAC) <35, 2010, Munich>
Englisch
Konferenzbeitrag
Fraunhofer SCAI ()

Abstract
We study non-holonomic overideals of a left differential ideal J ( F[8x, 8y] 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.

: http://publica.fraunhofer.de/dokumente/N-137402.html