Effective Power Series Computations

Let K be an effective field of characteristic zero. An effective tribe is a subset of K[[z₁, z₂, …]] = K∪ K[[z₁]] ∪ K[[z₁, z₂]] ∪ ⋯ that is effectively stable under the K-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.