Holomorphic Factorization of Matrices of Polynomials

  • John P. D’Angelo
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)


This paper considers some work done by the author and Catlin [CD1,CD2,CD3] concerning positivity conditions for bihomogeneous polynomials and metrics on bundles over certain complex manifolds. It presents a simpler proof of a special case of the main result in [CD3], providing also a self-contained proof of a generalization of the main result from [CD1]. Some new examples and applications appear here as well. The idea is to use the Bergman kernel function and some operator theory to prove purely algebraic theorems about matrices of polynomials.


Homogeneous Polynomial Bergman Kernel Principal Symbol Bergman Projection Proper Holomorphic Mapping 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [CD1].
    David W. Catlin and John P. D’Angelo, A stabilization theorem for Hermitian forms and applications to holomorphic mappings, Math Research Letters 3 (1996), 149–166.MathSciNetCrossRefMATHGoogle Scholar
  2. [CD2].
    David W. Catlin and John P. D’Angelo, Positivity conditions for bihomogeneous polynomials, Math Research Letters 4 (1997), 1–13.MathSciNetCrossRefGoogle Scholar
  3. [CD3].
    David W. Catlin and John P. D’Angelo, An isometric embedding theorem for holomorphic bundles. (preprint)Google Scholar
  4. [D].
    John P. D’Angelo, Several complex variables and the geometry of real hypersurf aces, CRC Press, Boca Raton (1993).Google Scholar
  5. [Dj].
    D. Z. Djokovic, Hermitian matrices over polynomial rings, J.Algebra 43 (1976), 359–374.MathSciNetCrossRefMATHGoogle Scholar
  6. [F].
    Gerald B. Folland, Introduction to Partial Differential Equations, Princeton University Press (1976).Google Scholar
  7. [RR].
    M. Rosenblum and J. Rovnyak, The factorization problem for nonnegative operator valued functions, Bulletin A.M.S. 77 (1971), 287–318.MathSciNetCrossRefMATHGoogle Scholar
  8. [Ru].
    Walter Rudin, Functional Analysis, McGraw-Hill, New York (1973).MATHGoogle Scholar
  9. [W].
    Raymond O. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall, Englewood Cliffs, New Jersey (1973).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • John P. D’Angelo
    • 1
  1. 1.Dept. of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations