# Resúmenes - Abstracts

Maria Emilia Alonso

 ALGEBRAIC POWER SERIES: REPRESENTATION AND EFFECTIVENESS OF DIVISION 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.

Robert Corless

 POLYNOMIAL ALGEBRA BY VALUES This talk outlines some new algorithms for operations on multivariate polynomials. The point of view taken in the talk is a dual viewpoint, namely that all polynomials considered are given by their values at known points, and that the degrees of the polynomials are known or can be deduced. The first nontrivial algorithm considered is division; then Bezout matrices, generalized companion matricies, GCD, and the solution of zero-dimensional polynomial systems are investigated. A new algorithm is given for such solution: neither Gröbner basis nor resultant-based, it shares some characteristics with resultant algorithms. A preprint of a paper detailing this talk can be found at http://www.orcca.on.ca/TechReports/2004/TR-04-01.html. This is joint work with Laureano Gonzalez-Vega, Amir Amiraslani, and Azar Shakoori.

Michel Coste

 CLASSIFYING SERIAL MANIPULATORS: COMPUTER ALGEBRA AND GEOMETRIC INSIGHT. This talk is intended to present the work of a group including researchers in robotics, computer algebra and geometry and studying the possible behaviours of a simple manipulator when one varies the design parameters (length of arms, offset). One important aspect of this behaviour is the ability to change posture without crossing singularities. The output of the work is a partition of the space of parameters into cells where the behaviour remains the same. We plan to emphasize the interplay between the algebraic modelisation of the problem, the use of specialized computer algebra tools and the geometric analysis of occurring phenomena.

David Cox

 IMPLICITATION AND COMMUTATIVE ALGEBRA The talk would discuss implicitization using A) resultants and B) moving surfaces and C) relations between the two. The talk would illustrate how concrete questions in implicitization leads naturally to concepts in commutative algebra such as syzygies, free resolutions, regularity, and local complete intersections. The talk would be down-to-earth but the moral would be that serious commutative algebra is involved. There would be lots of examples.

Peter Olver

 COMPUTATIONAL ASPECTS OF MOVING FRAMES In this talk, I will describe a new approach to the powerful Cartan theory of moving frames. The method is completely algorithmic, and applies to very general Lie group actions. I hope to discuss a wide variety of new applications, including classification of differential invariants, geometrical equivalence of curves and surfaces, symmetries of polynomials in classical invariant theory, object recognition in computer vision, invariant variational problems, and the design of symmetry-preserving numerical approximations. the emphasis will be on computer algebra aspects of the methods.

José Luis Ruiz Reina