Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 

 

The location of critical points of finite Blaschke products


Author: David A. Singer
Journal: Conform. Geom. Dyn. 10 (2006), 117-124
MSC (2000): Primary 53A35; Secondary 30D50
Published electronically: June 7, 2006
MathSciNet review: 2223044
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A theorem of Bôcher and Grace states that the critical points of a cubic polynomial are the foci of an ellipse tangent to the sides of the triangle joining the zeros. A more general result of Siebert and others states that the critical points of a polynomial of degree $ N$ are the algebraic foci of a curve of class $ N-1$ which is tangent to the lines joining pairs of zeroes. We prove the analogous results for hyperbolic polynomials, that is, for Blaschke products with $ N$ roots in the unit disc.


References [Enhancements On Off] (What's this?)

  • 1. Maxime Bôcher, Some propositions concerning the geometric representation of imaginaries, Ann. of Math. 7 (1892/93), no. 1-5, 70–72. MR 1502144, 10.2307/1967882
  • 2. Ulrich Daepp, Pamela Gorkin, and Raymond Mortini, Ellipses and finite Blaschke products, Amer. Math. Monthly 109 (2002), no. 9, 785–795. MR 1933701, 10.2307/3072367
  • 3. J.H. Grace, The zeros of a polynomial, Proc. Cambridge Philos. Soc. 11 (1902), 352-357.
  • 4. H. Hilton, Plane Algebraic Curves, second edition, London, Oxford University Press, 1932.
  • 5. F. Lucas, Propriétés géométriques des fractions rationelles, Paris Comptes Rendus 78 (1874), 271-274.
  • 6. Morris Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, Mathematical Surveys, No. 3, American Mathematical Society, New York, N. Y., 1949. MR 0031114
  • 7. J. Milnor, How to Compute Volume in Hyperbolic Space, in John Milnor, Collected Papers, Publish or Perish, Inc., Houston, 1994.
  • 8. Boris Mirman and Pradeep Shukla, A characterization of complex plane Poncelet curves, Linear Algebra Appl. 408 (2005), 86–119. MR 2166857, 10.1016/j.laa.2005.05.016
  • 9. George Salmon, A Treatise on the Higher Plane Curves, Third Edition, G. E. Stechert & Co., New York, 1934.
  • 10. Friedrich Schilling, Die Brennpunktseigenschaften der eigentlichen Ellipse in der ebenen nichteuklidischen hyperbolischen Geometrie, Math. Ann. 121 (1950), 415–426 (German). MR 0035035
  • 11. J. Siebeck, Ueber eine neue analytische Behandlungweise der Brennpunkte, J. Reine. Angew. Math. 64 (1864), 175.
  • 12. D. Singer, Critical Points of Hyperbolic Cubic Polynomials.
  • 13. C. E. Springer, Geometry and analysis of projective spaces, W. H. Freeman and Co., San Francisco, Calif.-London, 1964. MR 0173183
  • 14. Serge Tabachnikov, Dual billiards in the hyperbolic plane, Nonlinearity 15 (2002), no. 4, 1051–1072. MR 1912286, 10.1088/0951-7715/15/4/305
  • 15. Alexander P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys. 7 (1990), no. 1, 81–107. MR 1094732, 10.1016/0393-0440(90)90021-T
  • 16. J. L. Walsh, Note on the location of zeros of extremal polynomials in the non-euclidean plane, Acad. Serbe Sci. Publ. Inst. Math. 4 (1952), 157–160. MR 0049385

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 53A35, 30D50

Retrieve articles in all journals with MSC (2000): 53A35, 30D50


Additional Information

David A. Singer
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058
Email: david.singer@case.edu

DOI: https://doi.org/10.1090/S1088-4173-06-00145-7
Received by editor(s): January 16, 2006
Published electronically: June 7, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.