{
  "abstract": "Fix a polynomial $P(x)=\\sum_{j=0}^m a_jx^j$ of degree $m$, and let $I_kP$ be its $k$-fold antiderivative with all integration constants zero. Borwein, Chen, and Dilcher proved that, after $x=ky$, the nonzero zeros of $I_kP$ converge to the nonzero zeros of $L[P](y)=\\sum_j a_j j! y^j$. That convergence determines the leading linear motion but neither the constant offset nor successive differences. We compute the next two terms at a simple nonzero zero $λ$ of $L[P]$. Integer-order and Riemann-Liouville integration reduce, after scaling, to one rational family $c_j(h)=\\prod_{\\ell\\le j}(1+\\ell h)^{-1}$ sampled at $h=1/k$ and $h=1/α$. We prove $r_k=kλ+C_P(λ)+D_P(λ)/k+O(k^{-2})$ with explicit $C_P$ and $D_P$. Hence $λ$ is at once the scaled zero, the limit of successive root differences, and the continuous-order velocity, each with an explicit correction; $C_P(λ)$ is the offset of an asymptotic line. For binomials the limiting spacing is a multiple of a gamma-ratio quantity, the Factorial Fraction, and the nonzero reduced zeros form an exact regular polygon at every positive order. For real quadratics the collision order of two branches takes a simple discriminant form. In every degree the collision discriminant carries a universal factor, the one familiar from Laguerre polynomials and supported at negative orders. In the coordinates $(α,x)$ all trajectories lie on one plane curve of degree $m$ with leading form $L[P]$. At a nonzero zero of $L[P]$ of multiplicity $q$, the nearby zeros split on the scale $\\sqrt{k}$ like the zeros of the Hermite polynomial $He_q$, and for real $P$ the nonzero zeros stay real at every positive order exactly when $L[P]$ is real-rooted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12498",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick Lawless"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eay6zgqhwnqbr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12498",
  "fulltext_url": null,
  "id": "eay6zgqhwnqbr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12498",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7221508281",
  "pdf_url": "https://arxiv.org/pdf/2610.12498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eay6zgqhwnqbr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.12498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Asymptotic Root Dynamics under Repeated and Fractional Integration",
  "title_text": "Asymptotic Root Dynamics under Repeated and Fractional Integration",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/eay6zgqhwnqbr",
  "venue": "arXiv",
  "versions": []
}