Auteur / Autrice : | Pierre-Jean Spaenlehauer |
Direction : | Jean-Charles Faugère |
Type : | Thèse de doctorat |
Discipline(s) : | Informatique |
Date : | Soutenance en 2012 |
Etablissement(s) : | Paris 6 |
Mots clés
Mots clés contrôlés
Résumé
Multivariate polynomial systems arising in Engineering Science often carryalgebraic structures related to the problems they stem from. Inparticular, multi-homogeneous, determinantal structures and booleansystems can be met in a wide range of applications. A classical method to solve polynomial systems is to compute a Gröbner basis ofthe ideal associated to the system. This thesis provides new tools forsolving such structured systems in the context of Gröbner basis algorithms. On the one hand, these tools bring forth new bounds on the complexity of thecomputation of Gröbner bases of several families of structured systems(bilinear systems, determinantal systems, critical point systems,boolean systems). In particular, it allows the identification of families ofsystems for which the complexity of the computation is polynomial inthe number of solutions. On the other hand, this thesis provides new algorithms which takeprofit of these algebraic structures for improving the efficiency ofthe Gröbner basis computation and of the whole solving process(multi-homogeneous systems, boolean systems). These results areillustrated by applications in cryptology (cryptanalysis of MinRank),in optimization and in effective real geometry (critical pointsystems)