论文标题
Müger和Tuset关于多项式矩的结果的评论
Remarks on results by Müger and Tuset on the moments of polynomials
论文作者
论文摘要
令$ f（x）$为具有复杂系数的非零多项式，而$ m_p = \ int_ {0}^1 f（x）^p dx $ for $ p $ a正整数。在最近的一篇论文中，Müger和Tuset表明$ \ limsup_ {p \ to \ infty} | m_p |^{1/p}> 0 $，并指出此限制等于$ f $的最大值，以及$ f $的最大值，以及值$ | f（0）| $和$ | $和$ | f（1）| $。我们举了一个示例，表明此猜想是错误的。猜测$ \ limsup_ {p \ to \ infty} | m_p |^{1/p} $也很自然地等于$ | f（x）| $ [0,1] $。但是，我们也对此进行反例。我们还提供了有关数量$ \ limsup_ {p \ to \ infty} | m_p |^{1/p} $的行为的更多猜测。
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 $\limsup_{p \to \infty} |M_p|^{1/p}$.
