Exterior power sums

Abstract

We prove that for every fixed $\hat{I}»>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\ge 1$ satisfy ${\max }_{2\le k\le n+1}|{\sum }_j{z}_j^k|>e^{-\hat{I}»n}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erdős 973 can exist. The proof combines a truncated exponential factorization with overconvergence on an open set outside the unit disk and a normal-family obstruction for Cauchy transforms.

7 pages, 0 figures