Abstract
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 n F →A n F , and show that if f is birational, then ord f −1≤n 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.
Partially supported by GDR G0060 Calcul Formel, Algorithmes, Langages et Systèmes and PRC Mathématiques et Informatique.
Équipe Algèbre et Géométrie Algorithmiques, Calcul Formel, SDI CNRS no 6176 et Centre de Mathématiques, Unité de Recherche Associée au CNRS no D0169
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Ollivier, F. (1991). Standard bases of differential ideals. In: Sakata, S. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1990. Lecture Notes in Computer Science, vol 508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54195-0_60
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DOI: https://doi.org/10.1007/3-540-54195-0_60
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