[
  {
    "id": "ARR-2026-6TQRG7PY6C8JZR2D-v1",
    "type": "article",
    "title": "A sharp circle-root residual barrier and its collapse with one interior zero",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          21
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-6TQRG7PY6C8JZR2D",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-6TQRG7PY6C8JZR2D/versions/v1/",
    "note": "Research preprint; ARR-2026-6TQRG7PY6C8JZR2D v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "For degree-$n$ polynomials with all zeros on the unit circle, the least normalized residual $\\|P'-\\lambda P\\|_{H^2}/\\|P\\|_{H^2}$ tends to $1/\\sqrt8$, both for arbitrary complex $\\lambda$ and for the restricted slope $\\lambda=P'(0)$, $P(0)=1$. We prove the sharp lower limit through the classical Mathieu ground state and give an explicit circle-root family attaining it in every large degree. In contrast, exactly one zero strictly inside the disk, with all remaining zeros on the circle, permits restricted residual $O(n^{-1/6})$, giving the exact zero-count threshold for collapse. More generally, for each fixed $A>0$, a fixed number $M_A$ of interior zeros permits $O(n^{-A})$. Their distance and radial gap from 1 have order $n^{-1/3}$; explicit algebraic parameters in the one-root case, and an implicit-function construction in general, correct both degree rounding and the exact slope. For all disk-root polynomials and arbitrary slope, we also prove a uniform lower bound $\\exp(-C\\sqrt n)$ for every $C>\\sqrt{(8/3)\\log17}$, explicitly $\\exp(-3\\sqrt n)$ for $n\\ge10^4$. This refines the known residual method for exterior power sums; the negative answer to Erdos Problem 973 was already known. The optimal disk and one-root decay rates, and small exterior power-sum constructions, remain unresolved."
  }
]