Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 


A bound for the degree of a system of equations determining the variety of reducible polynomials

Author: A. L. Chistov
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 513-528
MSC (2010): Primary 14Q15; Secondary 14M99, 12Y05, 12E05, 13P05
Published electronically: March 21, 2013
MathSciNet review: 3014133
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathbb{A}}^N(\bar {K})$ denote the affine space of homogeneous polynomials of degree $ d$ in $ n+1$ variables with coefficients from the algebraic closure $ \bar {K}$ of a field $ K$ of arbitrary characteristic; so $ N=\binom {n+d}{n}$. It is proved that the variety of all reducible polynomials in this affine space can be given by a system of polynomial equations of degree less than $ 56d^7$ in $ N$ variables. This result makes it possible to formulate an efficient version of the first Bertini theorem for the case of a hypersurface.

References [Enhancements On Off] (What's this?)

  • 1. N. Bourbaki, Commutative algebra. Ch. 1-7, Springer-Verlag, Berlin, 1998. MR 1727221 (2001g:13001)
  • 2. A. L. Chistov, An algorithm of polynomial complexity for factoring polynomials, and determination of the components of a variety in a subexponential time, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 137 (1984), 124-188; English transl. in J. Soviet Math. 34 (1986), no. 4. MR 762101 (86g:11077b)
  • 3. -, Double-exponential lower bound for the degree of any system of generators of a polynomial prime ideal, Algebra i Analiz 20 (2008), no. 6, 186-213; English transl., St. Petersburg Math. J. 20 (2009), no. 6, 983-1001. MR 2530898 (2010e:13015)
  • 4. -, A deterministic polynomial-time algorithm for the first Bertini theorem, Preprint of St. Petersburg Math. Soc., 2004,
  • 5. A. L. Chistov, H. Fournier, L. Gurvits, and P. Koiran, Vandermonde matrices, NP-completeness, and transversal subspaces, Found. Comput. Math. 3 (2003), no. 4, 421-427. MR 2009684 (2004h:15026)
  • 6. A. K. Lenstra, H. W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534. MR 682664 (84a:12002)
  • 7. O. Zariski, Pencils on an algebraic variety and a new proof of a theorem of Bertini, Trans. Amer. Math. Soc. 50 (1941), 48-70. MR 0004241 (2:345a)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 14Q15, 14M99, 12Y05, 12E05, 13P05

Retrieve articles in all journals with MSC (2010): 14Q15, 14M99, 12Y05, 12E05, 13P05

Additional Information

A. L. Chistov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, Saint Petersburg 191023, Russia

Keywords: Absolute irreducibility, lattices, Bertini theorem
Received by editor(s): November 1, 2011
Published electronically: March 21, 2013
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society