Erdős Problem #973

Does there exist a constant \(C>1\) such that, for every \(n\geq 2\), there exists a sequence \(z_i\in \mathbb{C}\) with \(z_1=1\) and \(\lvert z_i\rvert \geq 1\) for all \(1\leq i\leq n\) with \(\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}\)? This is Problem 7.3 in [Ha74], where it is attributed to Erdős. The answer is no, by Luo, Yang and Zhu [LYZ26] (and via the residual-polynomial method of Tan, Wang, Huang and Chen [TWHC26]): the maximum exceeds \(e^{-\lambda n}\) for every fixed \(\lambda>0\) once \(n\) is large, so it decays subexponentially and no such \(C\) exists. See `erdos_973.variants.luo_yang_zhu` below.