关于Müger和Tuset关于多项式矩结果的注记
Remarks on results by Müger and Tuset on the moments of polynomials
摘要：
令$f(x)$为具有复系数的非零多项式，并且对于$p$，$M_p = \int_{0}^1 f(x)^p dx$为正整数.在最近的一篇论文中，Müger和Tuset证明了$\limsup_{p \to \infty} |M_p|^{1/p} > 0$，并阐明该极限等于$f$临界值中的最大值以及$|f(0)|$和$|f(1)|$。我们给出了一个例子，表明这个猜想是错误的。猜测$\limsup_{p \to \infty} |M_p|^{1/p}$等于$[0,1]$上的$|f(x)|$的最大值也是自然的。然而，我们也给出了一个反例。我们还提供了一些关于量$\limsup_{p \to \infty} |M_p|^{1/p}$的行为的猜测。
英文摘要：
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 $\limsup_{p \to \infty} |M_p|^{1/p} > 0$, 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 $\limsup_{p \to \infty} |M_p|^{1/p}$.
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