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
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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.
@article{ZNSL_2013_411_a11,
author = {A. L. Chistov},
title = {A deterministic polynomial-time algorithm for the first {Bertini} {theorem.~I}},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {191--239},
publisher = {mathdoc},
volume = {411},
year = {2013},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a11/}
}
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/