Geodesic
Polynomial-time computation of the degree of a~dominant morphism in characteristic zero.~I
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 189-235

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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. 
@article{ZNSL_2004_307_a6,
     author = {A. L. Chistov},
     title = {Polynomial-time computation of the degree of a~dominant morphism in characteristic {zero.~I}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {189--235},
     publisher = {mathdoc},
     volume = {307},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_307_a6/}
}
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A. L. Chistov. Polynomial-time computation of the degree of a~dominant morphism in characteristic zero.~I. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 189-235. http://geodesic.mathdoc.fr/item/ZNSL_2004_307_a6/