Geodesic
Polynomial-time algorithms for a~new model of representation of algebraic varieties (in characteristic zero)
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVIII, Tome 378 (2010), pp. 133-170

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We suggest a model of representation of algebraic varieties based on representative systems of points of their irreducible components. Deterministic polynomial-time algorithms that substantiate this model are described in characteristic zero. The main result here is a construction of the intersection of algebraic varieties. As applications, we obtain efficient algorithms for constructing the smooth stratification and smooth cover of an algebraic variety suggested by the author earlier. Bibl. 16 titles. 
@article{ZNSL_2010_378_a9,
     author = {A. L. Chistov},
     title = {Polynomial-time algorithms for a~new model of representation of algebraic varieties (in characteristic zero)},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {133--170},
     publisher = {mathdoc},
     volume = {378},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_378_a9/}
}
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A. L. Chistov. Polynomial-time algorithms for a~new model of representation of algebraic varieties (in characteristic zero). Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVIII, Tome 378 (2010), pp. 133-170. http://geodesic.mathdoc.fr/item/ZNSL_2010_378_a9/