[
  {
    "id": "ARR-2026-5286KSE6RS984S9P-v1",
    "type": "article",
    "title": "Interior-zero complexity of polynomial differentiation residuals",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          21
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-5286KSE6RS984S9P",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-5286KSE6RS984S9P/versions/v1/",
    "note": "Research preprint; ARR-2026-5286KSE6RS984S9P v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "We classify the numbers of interior zeros that permit vanishing and superalgebraic differentiation residuals for polynomials whose zeros lie in the closed unit disk. If a degree-$n$ polynomial has exactly $M$ strictly interior zeros, then, for every complex slope, its normalized $H^2$ residual is at least $n/[2\\sqrt216^M(n+2M+4)^{2M+2}]$. Consequently a prescribed budget $b_n$ permits a superalgebraically small residual sequence if and only if $b_n\\to\\infty$. Sufficiency holds with real coefficients, normalization at the origin, the slope fixed by that normalization, and exactly $b_n$ interior zeros. In contrast, one interior zero already permits residuals $O(n^{-1/6})$, with family-specific asymptotic constant $\\sqrt{15}/4^{4/3}$, whereas the circle-only infimum tends to $1/\\sqrt8$. For one interior zero we also prove a universal $c n^{-13/6}$ lower bound; the optimal rate remains open. The finite-budget obstruction combines a reciprocal identity with classical Mahler, truncated-shift and Poisson-kernel estimates, and extends to a stable factor times a self-inversive cofactor. For unrestricted interior counts we prove $\\exp(-C\\sqrt n)$ lower bounds for every $C>\\sqrt{(8/3)\\log17}$. The upper construction corrects both the exact slope and degree rounding at every large degree. The resulting refinement of exterior power-sum bounds uses a known residual method; the negative answer to Erdos Problem 973 was already known. No matching fixed-budget rates or stretched-exponential upper bounds are claimed."
  }
]