Numerical Decomposition of the Rank-Deficiency Set of a Matrix of Multivariate Polynomials

  • Daniel J. Bates
  • Jonathan D. Hauenstein
  • Chris Peterson
  • Andrew J. Sommese
Part of the Texts and Monographs in Symbolic Computation book series (TEXTSMONOGR)


Let A be a matrix whose entries are algebraic functions defined on a reduced quasi-projective algebraic set X, e.g. multivariate polynomials defined on X:= ℂ N . The sets \({\mathcal S}_k(A)\), consisting of xX where the rank of the matrix function A(x) is at most k, arise in a variety of contexts: for example, in the description of both the singular locus of an algebraic set and its fine structure; in the description of the degeneracy locus of maps between algebraic sets; and in the computation of the irreducible decomposition of the support of coherent algebraic sheaves, e.g. supports of finite modules over polynomial rings. In this article we present a numerical algorithm to compute the sets \({\mathcal S}_k(A)\) efficiently.


rank deficiency matrix of polynomials homotopy continuation irreducible components numerical algebraic geometry polynomial system Grassmannians 


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  1. 1.
    D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler, Bertini: Software for Numerical Algebraic Geometry. Available at
  2. 2.
    D.J. Bates, C. Peterson, A.J. Sommese, and C.W. Wampler, Numerical computation of the genus of an irreducible curve within an algebraic set, 2007 preprint.Google Scholar
  3. 3.
    M. Beltrametti and A.J. Sommese, The adjunction theory of complex projective varieties, Expositions in Mathematics 16, Walter De Gruyter, Berlin, 1995.Google Scholar
  4. 4.
    G. Björck and R. Fröberg, A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots, Journal of Symbolic Computation 12 (1991), 329–336.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Borel and R. Remmert, Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1961/1962), 429–439.MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Borel, Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.Google Scholar
  7. 7.
    CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra. Available at Scholar
  8. 8.
    B. Dayton and Z. Zeng, Computing the multiplicity structure in solving polynomial system, Proceedings of ISSAC 2005 (Beijing, China), 116–123, 2005.Google Scholar
  9. 9.
    M. Giusti and J. Heinz, Kronecker’s smart, little black boxes, in: R.A. DeVore, A. Iserles, and E. Süli (eds.), Foundations of Computational Mathematics, London Mathematical Society Lecture Note Series 284, Cambridge University Press, 2001, 69–104.Google Scholar
  10. 10.
    M. Giusti, G. Lecerf, and B. Salvy, A Gröbner free alternative for polynomial system solving, J. Complexity 17 (2001), 154–211.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005). Available at
  12. 12.
    P.A. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Library, Reprint of the 1978 original, John Wiley & Sons Inc., New York, 1994.Google Scholar
  13. 13.
    W.V.D. Hodge and D. Pedoe, Methods of algebraic geometry, Vol. II, Reprint of the 1952 original, Cambridge University Press, Cambridge, 1994.Google Scholar
  14. 14.
    G. Lecerf, Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers, J. Complexity 19 (2003), 564–596.MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Leykin, J. Verschelde, and A. Zhao, Newton’s method with deflation for isolated singularities of polynomial systems, Theoretical Computer Science 359 (2006), 111-122.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Y. Lu, D. Bates, A.J. Sommese, and C.W. Wampler, Finding all real points of a complex curve, in: A. Corso, J. Migliore, and C. Polini (eds.), Algebra, Geometry and Their Interactions, Contemporary Mathematics 448, American Mathematical Society, 2007, 183–205.Google Scholar
  17. 17.
    D.R. Grayson and M.E. Stillman, Macaulay 2, A software system for research in algebraic geometry. Available at
  18. 18.
    A.P. Morgan, A transformation to avoid solutions at infinity for polynomial systems, Applied Mathematics and Computation 18 (1986), 77–86.Google Scholar
  19. 19.
    D. Mumford, Varieties defined by quadratic equations, in: Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome 1970, 29–100.Google Scholar
  20. 20.
    T. Ojika, Modified deflation algorithm for the solution of singular problems, I. A system of nonlinear algebraic equations, J. Math. Anal. Appl. 123 (1987), 199–221.MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    T. Ojika, S. Watanabe, and T. Mitsui, Deflation algorithm for the multiple roots of a system of nonlinear equations, J. Math. Anal. Appl. 96 (1983), 463–479.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    A.J. Sommese, Holomorphic vector-fields on compact Kaehler manifolds, Math. Ann. 210 (1974), 75–82.MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    A.J. Sommese, J. Verschelde and C.W. Wampler, Numerical decomposition of the solution sets of polynomial systems into irreducible components, SIAM Journal on Numerical Analysis 38 (2001), 2022–2046.MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    A.J. Sommese, J. Verschelde and C.W. Wampler, Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components, in: C. Ciliberto, F. Hirzebruch, R. Miranda, and M. Teicher (eds.), Proceedings of the 2001 NATO Advance Research Conference on Applications of Algebraic Geometry to Coding Theory, Physics, and Computation (Eilat, Israel), NATO Science Series II: Mathematics, Physics, and Chemistry 36, Springer, 2001, 297–315.Google Scholar
  25. 25.
    A.J. Sommese, J. Verschelde and C.W. Wampler, Symmetric functions applied to decomposing solution sets of polynomial systems, SIAM Journal on Numerical Analysis 40 (2002), 2026–2046.MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    A.J. Sommese, J. Verschelde, and C.W. Wampler, A method for tracking singular paths with application to the numerical irreducible decomposition, in: M.C. Beltrametti, F. Catanese, C. Ciliberto, A. Lanteri, and C. Pedrini (eds.), Algebraic Geometry, a Volume in Memory of Paolo Francia, W. de Gruyter, 2002, 329–345.Google Scholar
  27. 27.
    A.J. Sommese, J. Verschelde, and C.W. Wampler, Numerical irreducible decomposition using PHCpack, in: M. Joswig and N. Takayama (eds.), Algebra, Geometry, and Software Systems, Springer-Verlag, 2003, 109–130.Google Scholar
  28. 28.
    A.J. Sommese, J. Verschelde, and C.W. Wampler, Homotopies for Intersecting Solution Components of Polynomial Systems, SIAM Journal on Numerical Analysis 42 (2004), 1552–1571.MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    A.J. Sommese, J. Verschelde, and C.W. Wampler, Solving Polynomial Systems Equation by Equation, in: A. Dickenstein, F.-O. Schreyer, and A.J. Sommese (eds.), Algorithms in Algebraic Geometry, IMA Volumes in Mathematics and Its Applications 146, Springer Verlag, 2007, 133-152.Google Scholar
  30. 30.
    A.J. Sommese and C.W. Wampler, The Numerical Solution to Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore, 2005.Google Scholar
  31. 31.
    A.J. Sommese and C.W. Wampler, Exceptional sets and fiber products, Foundations of Computational Mathematics 8 (2008), 171–196.MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Verschelde, Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation, ACM Transactions on Mathematical Software 25 (1999), 251–276. Available at Scholar

Copyright information

© Springer-Verlag Vienna 2009

Authors and Affiliations

  • Daniel J. Bates
    • 1
  • Jonathan D. Hauenstein
    • 2
  • Chris Peterson
    • 1
  • Andrew J. Sommese
    • 2
  1. 1.Department of MathematicsColorado State UniversityFort CollinsUSA
  2. 2.Department of MathematicsUniversity of Notre DameNotre DameUSA

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