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

Consider a projective algebraic variety $W$ that 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 over a field of zero characteristic. We show how to compute the degree of a dominant rational morphism from $W$ to $W'$ with $\dim W=\dim W'$. The morphism is given by homogeneous polynomials of degree $d'$. This algorithm is deterministic and polynomial in $(dd')^n$ and the size of input.