Strong version of the basic deciding algorithm for the existential theory of real fields

Let $U$ be a real algebraic variety in $n$-dimensional affine space which is a set of all zeroes of a family of polynomials of degrees less than $d$. In the case when $U$ is bounded (it is the main case) an algorithm of polynomial complexity is described for constructing a subset of $U$ with the number of elements bounded from above by $d^n$ which for every $s$ has a non–empty intersection with every cycle with coefficients from ${\mathbb Z}/2{\mathbb Z}$ of dimension $s$ of the closure of the set of smooth points of dimension $s$ of $U$.