Advertisement

Matrix Methods for Solving Algebraic Systems

  • Ioannis Z. Emiris

Abstract

The problem of computing all common zeros of a system of polynomials is of fundamental importance in a wide variety of scientific and engineering applications. This article surveys efficient methods based on the sparse resultant for computing all isolated solutions of an arbitrary system of n polynomials in n unknowns. In particular, we construct matrix formulae which yield nontrivial multiples of the resultant thus reducing root-finding to the eigendecomposition of a square matrix.

Keywords

Matrix Polynomial Polynomial System Integer Point Mixed Volume Newton Polytopes 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Canny, J., Emiris, I.Z. (2000): A subdivision-based algorithm for the sparse resultant. J. ACM. 47:417–451MathSciNetMATHCrossRefGoogle Scholar
  2. Canny, J., Pedersen, P. (1993): An algorithm for the Newton resultant. Technical Report 1394, Computer Science Department, Cornell University, Ithaca, New YorkGoogle Scholar
  3. Cox, D., Little, J., O’Shea, D. (1998): Using algebraic geometry. Springer, New York (Graduate Texts in Mathematics, vol. 185)MATHGoogle Scholar
  4. D’Andrea, C., Emiris, I.Z. (2000): Solving sparse degenerate algebraic systems. Submitted for publication. ftp://ftp-sop.inria.fr/saga/emiris/publis/DAEmOO.ps.gzGoogle Scholar
  5. Emiris, I.Z., Canny, J.F. (1995): Efficient incremental algorithms for the sparse resultant and the mixed volume. J. Symb. Comput. 20:117–149MathSciNetMATHCrossRefGoogle Scholar
  6. Emiris, I.Z., Mourrain, B. (1999): Computer algebra methods for studying and computing molecular conformations. Algorithmica, Special issue on algorithms for computational biology. 25:372–402MathSciNetGoogle Scholar
  7. Emiris, I.Z., Pan, V.Y. (1997): The structure of sparse resultant matrices. In: Proceedings of the ACM-SIGSAM International Symposium on Symbolic and Algebraic Computation, ISSAC’ 97, Maui, Hawaii. Association of Computing Machinery, New York, pp. 189–196Google Scholar
  8. Gao, T., Li, T.Y., Wang, X. (1999): Finding isolated zeros of polynomial systems in C n with stable mixed volumes. J. Symb. Comput. 28:187–211MathSciNetMATHCrossRefGoogle Scholar
  9. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V. (1994): Discriminants and resultants. Birkhüuser, BostonMATHCrossRefGoogle Scholar
  10. Havel, T.F., Hyberts, S., Najfeld, I. (1997): Recent advances in molecular distance geometry. In: Bioinformatics, Springer, Berlin (Lecture Notes in Computer Science, vol. 1278)Google Scholar
  11. Nikitopoulos, T.G. (1999): Matrix perturbation theory in distance geometry. Bachelor’s thesis. INRIA Sophia-Antipolis. www.inria.fr/saga/stages/arch.htmlGoogle Scholar
  12. Sturmfels, B. (1994): On the Newton polytope of the resultant. J. Algebr. Combinat. 3:207–236MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • Ioannis Z. Emiris

There are no affiliations available

Personalised recommendations