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Solving parametric polynomial systems. (English) Zbl 1156.14044
The authors present a new algorithm for solving constructible or semi-algebraic systems in the indeterminates \([U,X]\) where \(U=[U_1,\dots ,U_d]\) is the set of parameters and \(X=[X_{d+1},\dots,X_n]\) the set of unknowns. To this end, they study the characterization of open subsets in the parameter space over which the number of solutions is constant. What they define as the “discriminant variety with respect to a given projection of a basic constructible set” describes well the points where the projection is not regular. They show that this object is optimal in some sense and, in most cases, easy to compute. In the complex case, they develop an algorithm that avoids costly computations and works on a large class of systems, the so-called “well-behaved systems”, that includes most systems coming from applications. A more complicated picture arises in the real case, where they make use of other tools to compute and study this variety. In both cases, the algorithms are efficient and provide improvements in the solving of nontrivial systems.

14P99 Real algebraic and real-analytic geometry
68W30 Symbolic computation and algebraic computation
55R80 Discriminantal varieties and configuration spaces in algebraic topology
dpgb; RAGlib
Full Text: DOI
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