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

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

Abstract

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.

Keywords

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

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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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