Polynomial-time computation of the degree of a dominant morphism in zero characteristic. III

Consider a projective algebraic variety $W$ which is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than $d$ in $n+1$ variables in zero characteristic. Consider a dominant rational morphism from $W$ to $W'$ given by homogeneous polynomials of degree $d'$. We suggest algorithms for constructing objects in general position related to this morphism. These algorithms are deterministic and polynomial in $(dd')^n$ and the size of the input.