Details:
Title  Solving parametric polynomial systems  Author(s)  Daniel Lazard, Fabrice Rouillier  Type  Article in Journal  Abstract  We present a new algorithm for solving basic parametric constructible or semialgebraic systems of the form C = x ∈ C n , p 1 ( x ) = 0 , … , p s ( x ) = 0 , f 1 ( x ) ≠ 0 , … , f l ( x ) ≠ 0 or S = x ∈ R n , p 1 ( x ) = 0 , … , p s ( x ) = 0 , f 1 ( x ) > 0 , … , f l ( x ) > 0 , where p i , f i ∈ Q [ U , X ] , U = [ U 1 , … , U d ] is the set of parameters and X = [ X d + 1 , … , X n ] the set of unknowns. If Π U denotes the canonical projection onto the parameter’s space, solving C or S is reduced to the computation of submanifolds U ⊂ C d or U ⊂ R d such that ( Π U − 1 ( U ) ∩ C , Π U ) is an analytic covering of U (we say that U has the ( Π U , C ) covering property). This guarantees that the cardinality of Π U − 1 ( u ) ∩ C is constant on a neighborhood of u , that Π U − 1 ( U ) ∩ C is a finite collection of sheets and that Π U is a local diffeomorphism from each of these sheets onto U . We show that the complement in Π U ( C ) ¯ (the closure of Π U ( C ) for the usual topology of C n ) of the union of all the open subsets of Π U ( C ) ¯ which have the ( Π U , C )covering property is a Zariski closed set which is called the minimal discriminant variety of C w.r.t. Π U , denoted as W D . We propose an algorithm to compute W D efficiently. The variety W D can then be used to solve the parametric system C (resp. S ) as long as one can describe Π U ( C ) ¯ ∖ W D (resp. R d ∩ ( Π U ( C ) ¯ ∖ W D ) ). This can be done by using the critical points method or an “open” cylindrical algebraic decomposition.  Keywords  Computer algebra, Parametric polynomial system, Polynomial system, Solving, Discriminant variety, Algorithms, Semialgebraic set, Constructible set  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717107000132 
Language  English  Journal  Journal of Symbolic Computation  Volume  42  Number  6  Pages  636  667  Year  2007  Edition  0  Translation 
No  Refereed 
No 
