Abstract
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.
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References
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.
David W. Catlin and John P. D’Angelo, Positivity conditions for bihomogeneous polynomials, Math Research Letters 4 (1997), 1–13.
David W. Catlin and John P. D’Angelo, An isometric embedding theorem for holomorphic bundles. (preprint)
John P. D’Angelo, Several complex variables and the geometry of real hypersurf aces, CRC Press, Boca Raton (1993).
D. Z. Djokovic, Hermitian matrices over polynomial rings, J.Algebra 43 (1976), 359–374.
Gerald B. Folland, Introduction to Partial Differential Equations, Princeton University Press (1976).
M. Rosenblum and J. Rovnyak, The factorization problem for nonnegative operator valued functions, Bulletin A.M.S. 77 (1971), 287–318.
Walter Rudin, Functional Analysis, McGraw-Hill, New York (1973).
Raymond O. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall, Englewood Cliffs, New Jersey (1973).
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© 1999 Springer Science+Business Media Dordrecht
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D’Angelo, J.P. (1999). Holomorphic Factorization of Matrices of Polynomials. In: Saitoh, S., Alpay, D., Ball, J.A., Ohsawa, T. (eds) Reproducing Kernels and their Applications. International Society for Analysis, Applications and Computation, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2987-0_2
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DOI: https://doi.org/10.1007/978-1-4757-2987-0_2
Publisher Name: Springer, Boston, MA
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Online ISBN: 978-1-4757-2987-0
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