Residual bounds for Schur-stable polynomials

Abstract

Let $r_n$ be the infimum of \[ \frac{\lVert P'-P'(0)P\rVert_{H^2}}{\lVert P\rVert_{H^2}} \] over all degree-$n$ polynomials $P$ satisfying $P(0)=1$ whose zeros lie in the closed unit disk. We prove the quantitative residual bound \[ r_n\geq \exp\!\bigl(-(1+o(1))\sqrt n\log n\bigr) \qquad(n\to\infty). \] As an application, we answer Erdős Problem 973 on exterior power sums in the negative, in a form quantitatively stronger than the answer first obtained by Luo, Yang, and Zhu.

10 pages. Xiaojun Tan and Qihang Wang contributed equally. Substantially revised: the paper now centers on residual bounds for Schur-stable polynomials and proves r_n >= exp(-(1+o(1)) sqrt(n) log n). Erdos Problem 973 is presented as an application