Summary: [For the entire collection see Zbl 0727.00017.]
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 Guiseppa Carrá-Ferro, and following the general definition of standard bases, given in the author’s thesis (École Polytechnique 1990), 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 \(f:\mathbb{A}^ n_{\mathcal F}\mapsto \mathbb{A}^ n_{\mathcal F}\), and show that if \(f\) is birational, then \(\text{ord}(f^{-1})\leq n\cdot\text{ord}(f)\). 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 \(n\), \(\text{ord}(f)\) and the number of derivatives.