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Let $U$ be a real algebraic variety in $n$--dimensional affine space which is given as a set of zeros of a family of polynomials of the degree less than $d$. Let $U^{(s)}$ be the closure in the Zariski topology of set of all smooth points of dimension $s$ of $U$. In the paper an algorithm is described for constructing a set with a polynomial in $d^n$ number of points of $U$ which has a non--empty intersection with every connected component of dimension $s$ of every irreducible component of $U^{(s)}$. The similar result is valid for basic real semi--algebraic sets. More precise formulations are given involving triangulations of $U$ if $U$ is bounded (respectively the Alexandrov compactification of $U$ if $U$ is not bounded). The working time of the algorithm (for the case of algebraic varieties) is polynomial in the size of input and $d^n$.

Strong Version of the Basic Deciding Algorithm for the Existential Theory of Real Fields