computational complexity

, Volume 4, Issue 2, pp 107–132 | Cite as

Finding irreducible components of some real transcendental varieties

  • Marie-Françoise Roy
  • Nicolai Vorobjov
Article

Abstract

An algorithm is proposed for producing all components of the varieties defined by equations which involve polynomials and exponentials of polynomials, irreducible over real algebraic numbers. The running time of the algorithm is singly exponential in the number of variables and, with this number fixed, polynomial in all other parameters of the input.

Subject classifications

68Q25 

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References

  1. J. Bochnak, M. Coste, andM.-F. Roy,Géométrie Algébrique Réelle. Springer-Verlag, Berlin, 1987.Google Scholar
  2. A. L. Chistov, Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time. InZapiski Nauchnykh Seminarov LOMI 137 (1984), 124–188. English translation inJ. Soviet Math. 34 (1986), 1838–1882.Google Scholar
  3. M. Davis,Applied Nonstandard Analysis. J. Wiley, New York, 1977.Google Scholar
  4. A. Galligo and N. Vorobjov, Finding irreducible components of a real algebraic variety in subexponential time. Manuscript, 1992.Google Scholar
  5. D. Yu. Grigoriev, Factorization of polynomials over a finite field and the solution of systems of algebraic equations. InZapiski Nauchnykh Seminarov LOMI 137 (1984), 20–79. English translation inJ. Soviet Math. 34 (1986), 1762–1803.Google Scholar
  6. D. Yu. Grigoriev, The complexity of deciding Tarski algebra.J. Symbolic Comput. 5 (1988), 65–108.Google Scholar
  7. D. Yu. Grigoriev andN. Vorobjov, Solving systems of polynomial inequalities in subexponentiual time.J. Symbolic Comput. 5 (1988), 37–64.Google Scholar
  8. R. Hartshorn,Algebraic Geometry. Springer-Verlag, Berlin, 1977.Google Scholar
  9. J. Heintz, M.-F. Roy, andP. Solernó, Sur la complexité du principe de Tarski-Seidenberg.Bull. Soc. Math. France 118 (1990), 101–126.Google Scholar
  10. J. Heintz, M.-F. Roy, andP. Solernó, Single exponential path finding in semialgebraic sets II: The general case. InAlgebraic Geometry and its Applications, Springer-Verlag, Berlin, 1993, 467–481.Google Scholar
  11. E. Kaltofen, Computing the irreducible real factors and components of an algebraic curve.Appl. Alg. Eng. Comm. Comp. 1, (1990), 135–148.Google Scholar
  12. J. Renegar, On the computational complexity and geometry of the first order theory of reals.J. Symbolic Comput. 13 (1992), 329–352.Google Scholar
  13. I. R. Shafarevich,Basic Algebraic Geometry. Springer-Verlag, Berlin, 1974.Google Scholar
  14. N. Vorobjov, Deciding consistency of systems of polynomial in exponent inequalities in subexponential time. InEffective Methods in Algebraic Geometry, Birkhäuser, Boston, 1991, 491–501.Google Scholar
  15. N. Vorobjov, The complexity of deciding consistency of systems of polynomial in exponent inequalities.J. Symbolic Comput. 13 (1992), 139–173.Google Scholar
  16. H. Whitney, Elementary structure of real algebraic varieties.Ann. Math. 66 (1957), 456–467.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • Marie-Françoise Roy
    • 1
  • Nicolai Vorobjov
    • 2
  1. 1.I.R.M.A.R.Université de Rennes-1Rennes CedexFrance
  2. 2.Departments of Mathematics and Computer ScienceThe Pennsylvania State UniversityUniversity ParkUSA

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