Let be an effective field of characteristic zero. An effective tribe is a subset of which is effectively stable under the -algebra operations, restricted division, composition, the implicit function theorem, as well as restricted monomial transformations with arbitrary rational exponents. Given an effective tribe with an effective zero test, we will prove that an effective version of the Weierstrass division theorem holds inside the tribe, and that this can be used for the computation of standard bases.

Keywords: power series, algorithm, Weierstrass preparation, standard basis, D-algebraic power series, tribe

A.M.S. subject classification: 68W30, 03C60

Shorter 2014 version (no standard bases): Html, TeXmacs, Pdf, BibTeX

Latest 2016 version: Html, TeXmacs, Pdf, BibTeX