Asymptotics for Polynomial Zeros: Beware of Predictions from Plots

Abstract

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.

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Correspondence to Edward B. Saff.

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Dedicated to Walter Hayman on the occasion of his eightieth birthday

The research of E. B. Saff was supported, in part, by the U. S. National Science Foundation under grant DMS-060382.

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Saff, E.B., Stylianopoulos, N.S. Asymptotics for Polynomial Zeros: Beware of Predictions from Plots. Comput. Methods Funct. Theory 8, 385–407 (2008). https://doi.org/10.1007/BF03321695

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Keywords

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

2000 MSC

  • 30C10
  • 30C15
  • 30C40
  • 31A05
  • 31A15