Efficient construction of local parameters of irreducible components of an algebraic variety in nonzero characteristic

Consider an $(n-s)$-dimensional algebraic variety $W$ defined over an infinite field $k$ of nonzero characteristic $p$ and irreducible over this field. Let $W$ be a subvariety of the projective space of dimension $n$. We prove that the local ring of $W$ has a sequence of local parameters represented by $s$ nonhomogeneous polynomials with the product of degrees less than the degree of the variety multiplied by a constant depending on $n$. This allows us to prove the existence of effective smooth cover and smooth stratification of an algebraic variety in the case of ground field of nonzero characteristic. The paper extends the analogous results of the author obtained earlier in the case of zero characteristic of the ground field.