Efficient smooth stratification of an algebraic variety in zero characteristic and its applications

Let $V$ be an algebraic variety given by a system of homogeneous polynomials equations with degrees less than $d$ in $n+1$ variables. In zero-characteristic we prove that there is a smooth cover (smooth stratification) of $V$ with the number of strata at most $C(n)d^n$ (respectively $C(n)d^{n(n+1)/2}$) and degrees of strata at most $C(n)d^n$ where $C(n)>0$ depends only on $n$. Algorithms are suggested for constructing regular sequences and sequences of local parameters of irreducible components of $V$, computing dimension of a real algebraic variety with the complexity polynomial in $C(n)d^n$ and the size of input.