The location of critical points of finite Blaschke products
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- by David A. Singer
- Conform. Geom. Dyn. 10 (2006), 117-124
- DOI: https://doi.org/10.1090/S1088-4173-06-00145-7
- Published electronically: June 7, 2006
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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
- Maxime Bôcher, Some propositions concerning the geometric representation of imaginaries, Ann. of Math. 7 (1892/93), no. 1-5, 70–72. MR 1502144, DOI 10.2307/1967882
- Ulrich Daepp, Pamela Gorkin, and Raymond Mortini, Ellipses and finite Blaschke products, Amer. Math. Monthly 109 (2002), no. 9, 785–795. MR 1933701, DOI 10.2307/3072367 Gr J.H. Grace, The zeros of a polynomial, Proc. Cambridge Philos. Soc. 11 (1902), 352–357. Hi H. Hilton, Plane Algebraic Curves, second edition, London, Oxford University Press, 1932. Lu F. Lucas, Propriétés géométriques des fractions rationelles, Paris Comptes Rendus 78 (1874), 271–274.
- 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 Mil J. Milnor, How to Compute Volume in Hyperbolic Space, in John Milnor, Collected Papers, Publish or Perish, Inc., Houston, 1994.
- Boris Mirman and Pradeep Shukla, A characterization of complex plane Poncelet curves, Linear Algebra Appl. 408 (2005), 86–119. MR 2166857, DOI 10.1016/j.laa.2005.05.016 Sal George Salmon, A Treatise on the Higher Plane Curves, Third Edition, G. E. Stechert & Co., New York, 1934.
- Friedrich Schilling, Die Brennpunktseigenschaften der eigentlichen Ellipse in der ebenen nichteuklidischen hyperbolischen Geometrie, Math. Ann. 121 (1950), 415–426 (German). MR 35035, DOI 10.1007/BF01329635 Si J. Siebeck, Ueber eine neue analytische Behandlungweise der Brennpunkte, J. Reine. Angew. Math. 64 (1864), 175. Sin D. Singer, Critical Points of Hyperbolic Cubic Polynomials.
- C. E. Springer, Geometry and analysis of projective spaces, W. H. Freeman and Co., San Francisco, Calif.-London, 1964. MR 0173183
- Serge Tabachnikov, Dual billiards in the hyperbolic plane, Nonlinearity 15 (2002), no. 4, 1051–1072. MR 1912286, DOI 10.1088/0951-7715/15/4/305
- 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, DOI 10.1016/0393-0440(90)90021-T
- 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 49385
Bibliographic Information
- David A. Singer
- Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058
- Email: david.singer@case.edu
- Received by editor(s): January 16, 2006
- Published electronically: June 7, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn. 10 (2006), 117-124
- MSC (2000): Primary 53A35; Secondary 30D50
- DOI: https://doi.org/10.1090/S1088-4173-06-00145-7
- MathSciNet review: 2223044