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Journal of Mathematical Sciences

, Volume 200, Issue 6, pp 769–784 | Cite as

A Deterministic Polynomial-Time Algorithm for the First Bertini Theorem. II

  • A. L. Chistov
Article

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. This algorithm is deterministic and polynomial in (dd′)n and the size of the input. This paper is the second part in a tree-part series. Bibliography: 21 titles.

Keywords

Linear Form Irreducible Component Algebraic Variety Homogeneous Polynomial Implicit Function Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Baldassarry, Algebraic Varieties, Springer-Verlag, Berlin–Göttingen–Heidelberg (1956).CrossRefGoogle Scholar
  2. 2.
    N. Bourbaki, Algèbre commutative, Chaps. 1–7, Actualités Sci. Indust., nos. 1290, 1293, 1308, 1314, Paris (1961), (1964), (1965).Google Scholar
  3. 3.
    N. G. Chebotarev, Theory of Algebraic Functions [in Russian], OGIZ, Moscow–Leningrad (1948).Google Scholar
  4. 4.
    A. L. Chistov, “Polynomial complexity algorithm for factoring polynomials and constructing components of a variety in subexponential time,” J. Sov. Math., 34, No. 4, 1838–1882 (1986).CrossRefMATHGoogle Scholar
  5. 5.
    A. L. Chistov, “Polynomial-time computation of the degree of algebraic varieties in zero-characteristic and its applications,” J. Math. Sci., 108, No. 6, 897–933 (2002).CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. L. Chistov, “Efficient construction of local parameters of irreducible components of an algebraic variety,” Amer. Math. Soc. Transl. Ser. 2, 203, 201–231 (2001).MathSciNetGoogle Scholar
  7. 7.
    A. L. Chistov, “Strong version of the basic deciding algorithm for the existential theory of real fields,” J. Math. Sci., 107, No. 5, 4265–4295 (2001).CrossRefMathSciNetGoogle Scholar
  8. 8.
    A. L. Chistov, “Efficient smooth stratification of an algebraic variety in zero characteristic and its applications,” J. Math. Sci., 113, No. 5, 689–717 (2003).CrossRefMathSciNetGoogle Scholar
  9. 9.
    A. L. Chistov, “Monodromy and irreducibility criteria with algorithmic applications in zero characteristic,” J. Math. Sci., 126, No. 2, 1117–1127 (2005).CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. L. Chistov, “Polynomial-time computation of the degree of a dominant morphism in characteristic zero. I,” J. Math. Sci., 131, No. 2, 5547–5568 (2005).CrossRefMathSciNetGoogle Scholar
  11. 11.
    A. L. Chistov, “Polynomial-time computation of the degree of a dominant morphism in characteristic zero. II,” J. Math. Sci., 138, No. 3, 5733–5752 (2006).CrossRefMathSciNetGoogle Scholar
  12. 12.
    A. L. Chistov, “Polynomial-time computation of the degree of a dominant morphism in characteristic zero. III,” J. Math. Sci., 147, No. 6, 7234–7250 (2007).CrossRefMathSciNetGoogle Scholar
  13. 13.
    A. L. Chistov, “Polynomial-time computation of the degree of a dominant morphism in characteristic zero. IV,” J. Math. Sci., 158, No. 6, 912–927 (2009)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    A. L. Chistov, “A bound for the degree of a system of equations determining the variety of reducible polynomials,” Algebra Analiz, 24, No. 3, 199–222 (2012).MathSciNetGoogle Scholar
  15. 15.
    A. L. Chistov, “Polynomial-time computation of the dimensions of components of algebraic varieties in zero-characteristic,” J. Pure Appl. Algebra, 117 & 118, 145–175 (1997).CrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Chistov, H. Fournier, L. Gurvits, and P. Koiran, “Vandermonde matrices, NP-completeness, and transversal subspaces,” Found. Comput. Math., 3, No. 4, 421–427 (2003).CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    J. Bochnak, M. Coste, and M.-F. Roy, Geometrie algebrique réelle, Springer-Verlag, Berlin–Heidelberg–New York (1987).MATHGoogle Scholar
  18. 18.
    R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York–Heidelberg–Berlin (1977).CrossRefMATHGoogle Scholar
  19. 19.
    C. Jordan, Traité des substitutions et des équations algébriques, Paris (1870), pp. 277–279.Google Scholar
  20. 20.
    O. Zariski, “Pencils on an algebraic variety and a new proof of a theorem of Bertini,” Trans. Amer. Math. Soc., 50, 48–70 (1941).CrossRefMathSciNetGoogle Scholar
  21. 21.
    A. L. Chistov, “A deterministic polynomial-time algorithm for the first Bertini theorem. I,” J. Math. Sci., 196, No. 2, 223–243 (2014).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.St. Petersburg Department of Steklov Mathematical InstituteSt. PetersburgRussia

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