Distribution of simple zeros of polynomials

Abstract

1. I n t r o d u c t i o n a n d m a i n r e s u l t s It is well known that if Pn(x)=xn+... is a monic polynomial of degree n, then its supremum norm on [-1, 1] is at least as large as 21-n: 1 IIPnll[-~,~]/> 2n-----r' and here the equality sign holds only for the Chebyshev polynomials Tn(x) = 21-n cos(n arccos x). It is also known that if {Pn} is a sequence of monic polynomials with the property p , 1/n __ 1 l i m n i i l _ l l ] ~, n , } CCI