Standard bases of differential ideals

  • François Ollivier
Submitted Contributions Bases
Part of the Lecture Notes in Computer Science book series (LNCS, volume 508)


The aim of this paper is to introduce a new definition of standard bases of differential ideals, allowing more general orderings than the previous one, given by Giuseppa Carrá-Ferro, and following the general definition of standard bases, given in [O3], valid for algebraic ideals, canonical bases of subalgebras, etc.

Differential standard bases, as canonical bases, suffer a great limitation: they can be infinite, even for ideals of finite type. Nevertheless, we can sometimes bound the order of intermediate computations, necessary to make some elements of special interest appear in the basis.

As an illustration, we consider a differential rational map f: A F n →A F n , and show that if f is birational, then ord f−1n ord f. Partial standard bases computations provide then two algorithms to test the existence of f−1. The first one is also able to determine the inverse, if any. The second only determines existence, but we can provide a bound of complexity depending only of n, ord f and the number of derivatives.


Prime Ideal Standard Basis Canonical Basis Generic Zero Birational Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • François Ollivier
    • 1
  1. 1.Laboratoire d'Informatique de l'X (LIX)École PolytechniquePalaiseau CedexFrance

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