Computational Methods and Function Theory 8 (2008), No. 2, 385--407
Copyright Heldermann Verlag 2008
Edward B. Saff
edward.b.saff@Vanderbilt.Edu , Vanderbilt University, Department of Mathematics, 1326 Stevenson Center, 37240 Nashville, TN U.S.A.
Nikos S. Stylianopoulos
nikos@ucy.ac.cy , University of Cyprus, Department of Mathematics and Statistics, P.O. Box 20537, 1678 Nicosia, Cyprus.
We consider five plots of zeros corresponding to four eponymous planar polynomials (Szegö, 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.
Keywords: Szegö polynomials, Bergman polynomials, Faber polynomials, OPUC, zeros of polynomials, equilibrium measure.
MSC 2000: Primary 30C10; Secondary 30C15, 30C40, 31A05, 31A15.
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