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

Abstract

Let $f(x)$ be a non-zero polynomial with complex coefficients, and $M_p={\int }_0^{1}f(x{)}^{p}dx$ for $p$ a positive integer. In a recent paper, Müger and Tuset showed that ${\mathrm{limsup}}_{p\to \infty }|M_p{|}^{1/p}>0$, and conjectured that this limit is equal to the maximum amongst the critical values of $f$ together with the values $|f(0)|$ and $|f(1)|$. We give an example that shows that this conjecture is false. It also may be natural to guess that ${\mathrm{limsup}}_{p\to \infty }|M_p{|}^{1/p}$ is equal to the maximum of $|f(x)|$ 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 ${\mathrm{limsup}}_{p\to \infty }|M_p{|}^{1/p}$.

arXiv commentThe first version mischaracterized the conjecture by Müger and Tuset