Asymptotics for Polynomial Zeros: Beware of Predictions from Plots



We consider five plots of zeros corresponding to four eponymous planar polynomials (Szegő, Bergman, Faber and OPUC), for degrees up to 60, and state five conjectures suggested by these plots regarding their asymptotic distribution of zeros. By using recent results on zero distribution of polynomials we show that all these “natural” conjectures are false. Our main purpose is to provide the theoretical tools that explain, in each case, why these accurate, low degree plots are misleading in the asymptotic sense.


Szegő polynomials Bergman polynomials Faber polynomials OPUC zeros of polynomials equilibrium measure 

2000 MSC

30C10 30C15 30C40 31A05 31A15 


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  1. 1.
    V. V. Andrievskii and H.-P. Blatt, Erdős-Turán type theorems on quasiconformal curves and arcs, J. Approx. Theory 97 (1999) no.2, 334–365.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    V. V. Andrievskii and H.-P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics, Springer-Verlag, New York, 2002.MATHGoogle Scholar
  3. 3.
    V. V. Andrievskii and D. Gaier, Uniform convergence of Bieberbach polynomials in domains with piecewise quasianalytic boundary, Mitt. Math. Sem. Giessen (1992) no.211, 49–60.Google Scholar
  4. 4.
    V. V. Andrievskii and I. E. Pritsker, Convergence of Bieberbach polynomials in domains with interior cusps, J. Anal. Math. 82 (2000), 315–332.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    E. T. Copson, Partial Differential Equations, Cambridge University Press, Cambridge, 1975.MATHCrossRefGoogle Scholar
  6. 6.
    T. A. Driscoll, Algorithm 756: A matlab toolbox for Schwarz-Christoffel mapping, ACM Trans. Math. Soft. 22 (1996), 168–186.MATHCrossRefGoogle Scholar
  7. 7.
    M. Eiermann and H. Stahl, Zeros of orthogonal polynomials on regular n-gons, in: V. P. Havin and N. K. Nikolski (eds.), Linear and Complex Analysis Problem Book, V. 2, Lecture Notes in Mathematics, no. 1574, Springer-Verlag, 1994, pp. 187–189.Google Scholar
  8. 8.
    D. Gaier, Konstruktive Methoden der konformen Abbildung, Springer Tracts in Natural Philosophy, Vol. 3, Springer-Verlag, Berlin, 1964.Google Scholar
  9. 9.
    D. Gaier, On the decrease of Faber polynomials in domains with piecewise analytic boundary, Analysis (Munich) 21 (2001) no.2, 219–229.MathSciNetMATHGoogle Scholar
  10. 10.
    L. Golinskii, P. Nevai, F. Pintér, and W. Van Assche, Perturbation of orthogonal polynomials on an arc of the unit circle II, J. Approx. Theory 96 (1999) no.1, 1–32.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    M. X. He and E. B. Saff, The zeros of Faber polynomials for an m-cusped hypocycloid, J. Approx. Theory 78 (1994) no.3, 410–432.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    D. M. Hough, User’s Guide to CONFPACK, IPS Research Report 90-11, ETH-Zentrum, CH-8092 Zurich, Switzerland, 1990.Google Scholar
  13. 13.
    T. Kövari and Ch. Pommerenke, On Faber polynomials and Faber expansions, Math. Z. 99 (1967), 193–206.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    A. B. J. Kuijlaars and E. B. Saff, Asymptotic distribution of the zeros of Faber polynomials, Math. Proc. Cambridge Philos. Soc. 118 (1995) no.3, 437–447.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    A. L. Levin, E. B. Saff and N. S. Stylianopoulos, Zero distribution of Bergman orthogonal polynomials for certain planar domains, Constr. Approx. 19 (2003) no.3, 411–435.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    D. Levin, N. Papamichael and A. Sideridis, The Bergman kernel method for the numerical conformal mapping of simply connected domains, J. Inst. Math. Appl. 22 (1978) no.2, 171–187.MathSciNetMATHGoogle Scholar
  17. 17.
    A. Máté, P. Nevai and V. Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constr. Approx. 1 (1985) no.1, 63–69.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    V. Maymeskul and E. B. Saff, Zeros of polynomials orthogonal over regular N-gons, J. Approx. Theory 122 (2003) no.1, 129–140.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    V. V. Maymeskul, E. B. Saff and N. S. Stylianopoulos, L2-approximations of power and logarithmic functions with applications to numerical conformal mapping, Numer. Math. 91 (2002) no.3, 503–542.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    H. N. Mhaskar and E. B. Saff, On the distribution of zeros of polynomials orthogonal on the unit circle, J. Approx. Theory 63 (1990) no.1, 30–38.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    E. Miña-Díaz, E. B. Saff and N. S. Stylianopoulos, Zero distributions for polynomials orthogonal with weights over certain planar regions, Comput. Methods Funct. Theory 5 (2005) no.1, 185–221.MathSciNetMATHGoogle Scholar
  22. 22.
    N. Papamichael, E. B. Saff and J. Gong, symptotic behaviour of zeros of Bieberbach polynomials, J. Comput. Appl. Math. 34 (1991) no.3, 325–342.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    N. Papamichael and M. K. Warby, Stability and convergence properties of Bergman kernel methods for numerical conformal mapping, Numer. Math. 48 (1986) no.6, 639–669.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    I. E. Pritsker, Approximation of conformal mapping via the Szegő kernel method, Comput. Methods Funct. Theory 3 (2003) no.1-2, 79–94.MathSciNetMATHGoogle Scholar
  25. 25.
    E. A. Rakhmanov, The asymptotic behavior of the ratio of orthogonal polynomials II, Mat. Sb. (N.S.) 118(160) (1982) no.1, 104–117, 143.MathSciNetGoogle Scholar
  26. 26.
    T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995.Google Scholar
  27. 27.
    E. B. Saff, Polynomials of interpolation and approximation to meromorphic functions, Trans. Amer. Math. Soc. 143 (1969), 509–522.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, 1997.MATHGoogle Scholar
  29. 29.
    B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005, Classical theory.Google Scholar
  30. 30.
    B. Simon, Fine structure of the zeros of orthogonal polynomials, iii: periodic recursion coefficients, Comm. Pure Appl. Math. 59 (2006) no.7, 1042–1062.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    V. I. Smirnovand N. A. Lebedev, Functions of a Complex Variable: Constructive Theory, The M.I.T. Press, Cambridge, Mass., 1968.Google Scholar
  32. 32.
    H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.MATHCrossRefGoogle Scholar
  33. 33.
    P. K. Suetin, Series of Faber Polynomials, Analytical Methods and Special Functions, vol. 1, Gordon and Breach Science Publishers, Amsterdam, 1998, Translated from the 1984 Russian original by E. V. Pankratiev.MATHGoogle Scholar
  34. 34.
    G. Szegő, Orthogonal Polynomials, fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975.Google Scholar
  35. 35.
    J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, fourth edition, American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965.MATHGoogle Scholar

Copyright information

© Heldermann  Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA
  2. 2.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus

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