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

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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. 
@article{ZNSL_2012_403_a11,
     author = {A. L. Chistov},
     title = {An effective version of the first {Bertini} theorem in nonzero characteristic and its applications},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {172--196},
     publisher = {mathdoc},
     volume = {403},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a11/}
}
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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/