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We present a new probabilistic algorithm to find a finite set of pointsintersecting the closure of each connected component of the realization ofevery sign condition over a family of real polynomials defining regularhypersurfaces that intersect transversally. This enables us to show aprobabilistic procedure to list all feasible sign conditions over thepolynomials. In addition, we extend these results to the case of closed signconditions over an arbitrary family of real multivariate polynomials. Thecomplexity bounds for these procedures improve the known ones.

On sign conditions over real multivariate polynomials