On the moments of a polynomial in one variable (Q2288228)

The authors prove that for any non-zero polynomial with complex coefficients \(\lim \sup_{n\to \infty} |M_n(f)|^{1/n}>0\), where \(M_n(f)=\int_{0}^{1}( f(x))^n dx\). In particular, \(M_n(f) \neq 0\) for infinitely many \(n \in \mathbb {N}\). The last statement was proven earlier by \textit{J. P. Françoise} et al. [Bull. Sci. Math. 135, No. 1, 10--32 (2011; Zbl 1217.44008)] by mostly algebraic methods. Using some ideas of \textit{J. J. Duistermaat} and \textit{W. van der Kallen} [Indag. Math., New Ser. 9, No. 2, 221--231 (1998; Zbl 0916.22007)], the authors of the present paper provide an analytic proof of the result.