Resúmenes - Abstracts

In the talk we give an approach to some topics of Computational Algebra as standard bases computation and canonical forms in the setting of algebraic power series.
Our main motivation to deal with these problems comes from Algebraic Geometry, where Algebraic functions appear in
describing the local structure of algebraic varieties at a point.
There is a second more abstract motivation. The henselization of a local ring A is ``unique" in an strong way, (as for instance it is the real clousure of an ordered field); it is unique up to unique A--homomorphism. So that, in a more general setting we are dealing with describing, say constructively, the henselization of a local ring.
This is the approach where we show how we can manipulate the henselization of a local ring in a dynamic way through some essentially finite $A$ algebras.

The talk present results from joint works: old results, and new ones in progress with Castro and Hauser, and, Lombardi and Pedry. Our approach to compute in the ring of algebraic power series is the representation, based in the fact that an algebraic function locally is a component of a vector of solutions to a polynomial system of equations verifiying IFT at a point.

That representation allows to compute standard bases and the initial monomial ideal or the set of leading (minimal) terms E={ in(I) of a given ideal I of the ring of algebraic power series, wr.t. a local term ordering. We identify E=in(I) with a subset of Nn. By Hironaka Henselian Division Theorem it is known that quotients and remainders of the formal division by ( an standard basis of ) I are still algebraic series, provided E verifies a finiteness condition called ``box condition".
We achieve to have an effective version of this theorem, that is to effectively compute canonical forms, in some cases including Weierstrass division theorem.

For, we introduce a new reduction process of a polynomial w.r.t. a set of polynomials whose leading monomials generate an ``escalier" which has this finitness condition. In fact this process is very general and does not required to fix any term ordering for considering the leading (maximal) monomials of polynomials: one can fix freely the ``leading monomials" with the only condition to be maximal in the partial order of semigroup. We use old ideas of Janet to make a suitable partition of the set leading terms E.