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Polar varieties and efficient real elimination. (English) Zbl 1073.14554

Summary: Let \(S_0\) be a smooth and compact real variety given by a reduced regular sequence of polynomials \(f_1, \dots, f_p\). This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of \(S_0\). For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of \(S_0\). This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations \(f_1,\dots, f_p\) and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system \(f_1,\dots,f_p\).

MSC:

14P05 Real algebraic sets
14Q99 Computational aspects in algebraic geometry
68W30 Symbolic computation and algebraic computation
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