Remarks on results by M\"uger and Tuset on the moments of polynomials

TL;DR

This paper examines the asymptotic behavior of polynomial moments and provides counterexamples to previous conjectures relating these limits to critical values and maximum modulus of the polynomial.

Contribution

It disproves conjectures that the limit supremum of polynomial moments' p-th roots equals the maximum of critical values or the polynomial's maximum modulus on [0,1].

Findings

Counterexamples to M"uger and Tuset's conjecture.

Counterexamples to the conjecture relating to maximum modulus.

Proposed alternative hypotheses for the limit behavior.

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, M\"uger and Tuset showed that $\limsup_{p \to \infty} |M_p|^{1/p} > 0$, 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 $\limsup_{p \to \infty} |M_p|^{1/p}$ 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 $\limsup_{p \to \infty} |M_p|^{1/p}$.