Geodesic
A deterministic polynomial-time algorithm for the first Bertini theorem. I
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 191-239 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 degree less than $d$ in $n+1$ variables in zero characteristic. Consider a linear system on $W$ given by homogeneous polynomials of degree $d'$. Under the conditions of the first Bertini theorem for $W$ and this linear system, we show how to construct an irreducible divisor in general position from the statement of this theorem. The algorithm is deterministic and polynomial in $(dd')^n$ and the size of the input. 
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A. L. Chistov. A deterministic polynomial-time algorithm for the first Bertini theorem. I. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 191-239. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a11/

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