Geodesic
An effective version of the first Bertini theorem in nonzero characteristic and its applications
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 172-196 Cet article a éte moissonné depuis la source Math-Net.Ru

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We suggest an effective algorithm for the first Bertini theorem in nonzero characteristic of the ground field. This allows us to improve the double-exponential lower bound for the degree of any system of generators of a polynomial ideal over a finite ground field. 
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A. L. Chistov. An effective version of the first Bertini theorem in nonzero characteristic and its applications. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 172-196. http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a11/

[1] A. L. Chistov, “Algoritm polinomialnoi slozhnosti dlya razlozheniya mnogochlenov na neprivodimye mnozhiteli i nakhozhdenie komponent mnogoobraziya v subeksponentsialnoe vremya”, Zap. nauchn. semin. LOMI, 137, 1984, 124–188 | MR | Zbl

[2] A. L. Chistov, “Dvazhdy eksponentsialnaya nizhnyaya otsenka na stepen sistemy obrazuyuschikh polinomialnogo prostogo ideala”, Algebra i analiz, 20:6 (2008), 186–213 | MR

[3] A. L. Chistov, “Otsenka stepeni sistemy uravnenii, zadayuschei mnogoobrazie privodimykh mnogochlenov”, Algebra i analiz, 24:3 (2012), 199–222 | MR

[4] A. L. Chistov, “An improvement of the complexity bound for solving systems of polynomial equations”, Zap. nauchn. semin. POMI, 390, 2011, 299–306 | MR

[5] A. L. Chistov, A deterministic polynomial-time algorithm for the first Bertini theorem, Preprint, St. Petersburg Mathematical Society, 2004 http://www.mathsoc.spb.ru/preprint/2004/index.html