Gröbner bases specialization through Hilbert functions

Abstract

A b s t r a c tThis paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert function is used to check that the specialization of a 'generic' GrSbner basis of the parametric homogeneous polynomial system (computed in a polynomial ring containing the parameters and the unknowns as variables) is a Gr6bner basis of the specialized homogeneous polynomial system. A preliminary implementation of these algorithms in PoSSoLib is also reported.

Complexity of solving parametric polynomial systems

“…Dans ces deux stratégies on se ramèneà calculer un sous-ensemble constructible de P tel que la spécialisation des paramètres par des valeurs de cet ensemble conduità une base de Gröbner de l'idéal spécialisé [70,46,53,54,31,75]. Notons aussi les bases de Gröbner compréhensive [121,122] qui partagent l'espace des paramètres en des ensembles constructibles chacun avec une base de Gröbner paramétrique (pas de borne de complexité pour cette construction).…”

Improving the DISPGB algorithm using the discriminant ideal

“…[SaSu03] 8.2 10 Y Y (0) S14. [GoTrZa00,De99] 9.6 2 Y N (1) ROMIN robot S15. [GoRe93] 18.2 17 Y N (2) S16.…”

“…For S17 [GoTrZa00], DISPGB gets bogged down after computing 35 terminal vertices in 1375 sec. It has been unable to finish the tree, and so neither rebuilding with the discriminantideal nor reducing the tree can have been achieved.…”

An algorithm for solving zero-dimensional parametric systems of polynomial homogeneous equations

“…Such parametric polynomial systems come from real-life problems as geometric [12,25], optimization [41] and interpolation [35,36,15] ones, or physical problems [27,33,11], chemical reactions [10,11,15] and robots [16,6,35,36].…”