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Polynomials defining distinguished varieties


Author: Greg Knese
Journal: Trans. Amer. Math. Soc. 362 (2010), 5635-5655
MSC (2010): Primary 47A57, 47A13, 14M99, 32A10, 32A60, 14H50
DOI: https://doi.org/10.1090/S0002-9947-2010-05275-4
Published electronically: June 11, 2010
MathSciNet review: 2661491
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Abstract | References | Similar Articles | Additional Information

Abstract: Using a sums of squares formula for two-variable polynomials with no zeros on the bidisk, we are able to give a new proof of a representation formula for distinguished varieties. For distinguished varieties with no singularities on the two-torus, we are able to provide extra details about the representation formula and use this to prove a bounded extension theorem.


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  • 1. Kenzō Adachi, Mats Andersson, and Hong Rae Cho.
    $ L\sp p$ and $ H\sp p$ extensions of holomorphic functions from subvarieties of analytic polyhedra.
    Pacific J. Math., 189(2):201-210, 1999. MR 1696119 (2000c:32036)
  • 2. Jim Agler and John E. McCarthy.
    Distinguished varieties.
    Acta Math., 194(2):133-153, 2005. MR 2231339 (2007c:47006)
  • 3. Jim Agler and John E. McCarthy.
    Parametrizing distinguished varieties.
    In Recent advances in operator-related function theory, volume 393 of Contemp. Math., pages 29-34. Amer. Math. Soc., Providence, RI, 2006. MR 2198369 (2007i:32012)
  • 4. Jim Agler and John E. McCarthy.
    Hyperbolic algebraic and analytic curves.
    Indiana Univ. Math. J., 56(6):2899-2933, 2007. MR 2375707 (2009a:32039)
  • 5. Jim Agler, John E. McCarthy, and Mark Stankus.
    Toral algebraic sets and function theory on polydisks.
    J. Geom. Anal., 16(4):551-562, 2006. MR 2271943 (2007j:32002)
  • 6. Brian J. Cole and John Wermer.
    Ando's theorem and sums of squares.
    Indiana Univ. Math. J., 48(3):767-791, 1999. MR 1736979 (2000m:47014)
  • 7. Kenneth R. Davidson, Vern I. Paulsen, Mrinal Raghupathi, and Dinesh Singh.
    A Constrained Nevanlinna-Pick Interpolation Problem.
    Indiana Univ. Math. J. 58(2):709-732, 2009. MR 2514385
  • 8. S. I. Fedorov.
    Harmonic analysis in a multiply connected domain. I.
    Mat. Sb., 181(6):833-864, 1990. MR 1072300 (92m:30068a)
  • 9. Jeffrey S. Geronimo and Hugo J. Woerdeman.
    Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables.
    Ann. of Math. (2), 160(3):839-906, 2004. MR 2144970 (2006b:42036)
  • 10. Jeffrey S. Geronimo and Hugo J. Woerdeman.
    Two-variable polynomials: intersecting zeros and stability.
    IEEE Trans. Circuits Syst. I Regul. Pap., 53(5):1130-1139, 2006. MR 2235187 (2007c:93050)
  • 11. Greg Knese.
    Bernstein-Szegő measures on the two dimensional torus.
    Indiana Univ. Math. J. 57(3):1353-1376, 2008. MR 2429095 (2009h:46054)
  • 12. Greg Knese.
    Function theory on the Neil parabola.
    Michigan Math. J., 55(1):139-154, 2007. MR 2320176 (2008b:32004)
  • 13. Walter Rudin.
    Pairs of inner functions on finite Riemann surfaces.
    Trans. Amer. Math. Soc., 140:423-434, 1969. MR 0241629 (39:2968)
  • 14. Edgar Lee Stout.
    Bounded extensions. The case of discs in polydiscs.
    J. Analyse Math., 28:239-254, 1975. MR 0590066 (58:28662)
  • 15. Prasad Vegulla.
    Geometry of distinguished varieties.
    to appear in Michigan Math. J.

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Additional Information

Greg Knese
Affiliation: Department of Mathematics, University of California, Irvine, Irvine, California 92614-3875
Address at time of publication: Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350
Email: gknese@uci.edu, gknese@bama.ua.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05275-4
Keywords: Distinguished varieties, stable polynomials, bidisk
Received by editor(s): March 25, 2008
Published electronically: June 11, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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