Solving parametric polynomial systems

• Authors: Daniel Lazard;Fabrice Rouillier
• Venue: Journal of Symbolic Computation
• Year: 2007

We present a new algorithm for solving basic parametric constructible or semi-algebraic 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 @P"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 (@P"U^-^1(U)@?C,@P"U) is an analytic covering of U (we say that U has the (@P"U,C)-covering property). This guarantees that the cardinality of @P"U^-^1(u)@?C is constant on a neighborhood of u, that @P"U^-^1(U)@?C is a finite collection of sheets and that @P"U is a local diffeomorphism from each of these sheets onto U. We show that the complement in @P"U(C)@? (the closure of @P"U(C) for the usual topology of C^n) of the union of all the open subsets of @P"U(C)@? which have the (@P"U,C)-covering property is a Zariski closed set which is called the minimal discriminant variety ofCw.r.t.@P"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 @P"U(C)@?@?W"D (resp. R^d@?(@P"U(C)@?@?W"D)). This can be done by using the critical points method or an ''open'' cylindrical algebraic decomposition.