Geodesic
Effective construction of a nonsingular in codimension one algebraic variety over a zero-characteristic ground field
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 167-188 
A. L. Chistov. Effective construction of a nonsingular in codimension one algebraic variety over a zero-characteristic ground field. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 167-188. http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a7/
@article{ZNSL_2011_387_a7,
     author = {A. L. Chistov},
     title = {Effective construction of a~nonsingular in codimension one algebraic variety over a~zero-characteristic ground field},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--188},
     year = {2011},
     volume = {387},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a7/}
}
TY  - JOUR
AU  - A. L. Chistov
TI  - Effective construction of a nonsingular in codimension one algebraic variety over a zero-characteristic ground field
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 167
EP  - 188
VL  - 387
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a7/
LA  - en
ID  - ZNSL_2011_387_a7
ER  - 
%0 Journal Article
%A A. L. Chistov
%T Effective construction of a nonsingular in codimension one algebraic variety over a zero-characteristic ground field
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 167-188
%V 387
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a7/
%G en
%F ZNSL_2011_387_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $k$ be a field of zero-characteristic finitely generated over a primitive subfield. Let $f$ be a polynomial of degree at most $d$ in $n$ variables with coefficients from $k$ and irreducible over an algebraic closure $\overline k$. Then we construct a nonsingular in codimension one algebraic variety $V$ and a finite birational isomorphism $V\to\mathcal Z(f)$ where $\mathcal Z(f)$ is the hypersurface of all common zeroes of the polynomial $f$ in the affine space. The working time of the algorithm for constructing $V$ is polynomial in the size of the input.

[1] A. L. Chistov, “Polynomial complexity of the Newton–Puiseux algorithm”, Lecture Notes in Computer Science, 233, Springer, 1986, 247–255 | DOI | MR

[2] A. L. Chistov, “Polynomial-time algorithms for computational problems in the theory of algebraic curves”, Zap. nauchn. semin. POMI, 176, 1989, 127–150 | MR | Zbl

[3] A. L. Chistov, “An overview of effective normalization of a nonsingular in codimension one projective algebraic variety”, Zap. nauchn. semin. POMI, 373, 2009, 295–317 | MR

[4] A. L. Chistov, “Polynomial-time factoring polynomials over local fields”, Zap. nuchn. semin. POMI, 192, 1991, 112–148 | MR | Zbl

[5] A. L. Chistov, “The complexity of constructing the ring of integers of a global field”, Dokl. Akad. Nauk SSSR, 306:5 (1989), 1063–1067 | MR | Zbl

[6] A. L. Chistov, A deterministic polynomial-time algorithm for the first Bertini theorem, Preprint, St. Petersburg Mathematical Society, 2004 http://www.MathSoc.spb.ru | MR

[7] A. L. Chistov, “Effective Construction of a Nonsingular in Codimension One Algebraic Variety”, Polynomial Computer Algebra, Theses of Talks, 2010, 15–18

[8] A. L. Chistov, “Polynomial complexity algorithm for factoring polynomials and constructing components of a variety in subexponential time”, Zap. nauchn. semin. LOMI, 137, 1984, 124–188 | MR | Zbl