Remarks on results by Müger and Tuset on the moments of polynomials

Abstract

Let f (x) be a non-zero polynomial with complex coefficients, andp dx for p a positive integer. In a recent paper, Müger and Tuset showed that lim sup p!1 jMpj 1=p > 0, and conjectured that this limit is equal to the maximum amongst the critical values of f together with the values jf (0)j and jf (1)j. We give an example that shows that this conjecture is false. It also may be natural to guess that lim sup p!1 jMpj 1=p is equal to the maximum of jf (x)j on [0; 1]. However, we give a counterexample to this as well. We also provide a few more guesses as to the behaviour of the quantity lim sup p!1 jMpj 1=p .