LIP6 2001/008
- Thesis
Ensembles triangulaires de polynômes et résolution de systèmes algébriques - Ph. Aubry
- 171 pages - 01/13/1999- document en - http://www.lip6.fr/lip6/reports/2001/lip6.2001.008.pdf - 1,193 Ko
- Contact : Philippe.Aubry (at) nulllip6.fr
- Ancien Thème : CALFOR
- Keywords : computer algebra, triangular set, polynomial systems, Galois theory, Groebner, equidimensional decomposition
- Publisher : David.Massot (at) nulllip6.fr
In the first part of this thesis, we study the properties of differents notions of triangular polynomial sets and we present corresponding algorithms for the symbolic resolution of algebraic systems. We show in particular the pure equidimensionality of the saturate of a triangular set, and then deduce the equivalence of our simple notion of regular triangular set with the concept of regular chain. Our new algorithms allows more efficient triangular decompositions of polynomial systems and stronger specifications.
The second part treats constructive algebraic Galois theory by a new approach. We show with A. Valibouze that a Galois ideal of a separable polynomial f associated with a permutations group which contains the Galois group of f, is generated by a triangular Groebner basis. We develop new algorithms from this structure result (computation of relative resolvents and generators of Galois ideals).
The third part is devoted to implementation, experimentation and applications. We present a work of experimental comparison on four methods for computing triangular decompositions realized with M. Moreno Maza, then we give three applications of our implementations (Galois ideal of a polynomial, image compression, n body problem).
The second part treats constructive algebraic Galois theory by a new approach. We show with A. Valibouze that a Galois ideal of a separable polynomial f associated with a permutations group which contains the Galois group of f, is generated by a triangular Groebner basis. We develop new algorithms from this structure result (computation of relative resolvents and generators of Galois ideals).
The third part is devoted to implementation, experimentation and applications. We present a work of experimental comparison on four methods for computing triangular decompositions realized with M. Moreno Maza, then we give three applications of our implementations (Galois ideal of a polynomial, image compression, n body problem).